Finite group representations for the pure mathematician

Finite Group Representations for the Pure Mathematician by

Peter Webb

Preface This book started as notes for courses given at the graduate level at the University of Minnesota. It is intended to be used at the level of a second-year graduate course in an American university. At this level the book should provide material for a year-long course in representation theory. It is supposed that the reader has already studied the material in a first-year graduate course on algebra and is familiar with the basic properties, for example, of Sylow subgroups and solvable groups, as well as the examples which are introduced in a first group theory course, such as the dihedral, symmetric, alternating and quaternion groups. The reader should also be familiar with other aspects of algebra which appear in or before a firstyear graduate course, such as Galois Theory, tensor products, Noetherian properties of commutative rings, the structure of modules over a principal ideal domain, and the first properties of ideals. The Pure Mathematician for whom this course is intended may well have a primary interest in an area of pure mathematics other than the representation theory of finite groups. Group representations arise naturally in many areas, such as number theory, combinatorics and topology, to name just three, and the aim of this course is to give students in a wide range of areas the technique to understand the representations which they encounter. This point of view has determined to a large extent the nature of this book: it should be sufficiently short, so that students who are not specialists in group representations can get to the end of it. Since the representations which arise in many areas are defined over rings other than fields of characteristic zero, such as rings of algebraic integers or finite fields, the theory is developed over arbitrary ground rings where possible. The student finishing this course should feel no lack of confidence in working in characteristic p. A selection of topics has been made at every stage, and where a result appears to have predominantly technical interest, perhaps confined to those who specialize in group representations, then in some cases it has been omitted. The exercises at the ends of sections are an important part of this book. They provide a place to indicate how the subject may be developed beyond what is described in the text. They provide a stock of examples which provide a firmer basis for the expertize of the reader. And they provide homework exercises so that the reader can learn by actively doing, as well as by the more passive activity of reading.

Provisional Table of Contents

Part 1: Representations 1 2 3 4

5 6 7 8 9 10 11 12

Representation, Maschke’s theorem and semisimplicity The structure of algebras for which every module is semisimple Characters The construction of modules and characters Induction and restriction Symmetric and Exterior powers The construction of character tables More on induction and restriction: theorems of Mackey and Clifford Representations of p-groups in characteristic p Projective modules for finite-dimensional algebras Projective modules for group algebras Changing the ground ring: splitting fields and the decomposition map Brauer characters Indecomposable modules: vertices, sources and Green correspondence Blocks

Part 2: Group cohomology

1. Representations, Maschke’s Theorem and Semisimplicity We start by establishing some notation. We let G denote a finite group, and we will work over a commutative ring R with a 1. For example, R could be any of the fields Q, C, R, Fp , Fp etc, where the latter symbol denotes the algebraic closure of Fp , or we could take R = Z or some other ring. If V is an R-module we denote by GL(V ) the group of all invertible R-module homomorphisms V → V . In case V ∼ = Rn is a free module of rank n this group is isomorphic to the group of all non-singular n × n-matrices over R, and we denote it GL(n, R) or GLn (R), or in case R = Fq is the finite field with q elements by GL(n, q) or GLn (q). A (linear) representation of G (over R) is a homomorphism ρ : G → GL(V ). In a situation where V is free as an R-module, on taking a basis for V we obtain for each element g ∈ G a matrix ρ(G), and these matrices multiply together in the manner of the group. In this situation the rank of the free R-module V is called the degree of the representation. Sometimes by abuse of terminology the module V is called the representation, but it should more properly be called the representation module or representation space (if R is a field). (1.1) Examples. 1. For any group G and commutative ring R we can take V = R and ρ(G) = 1 for all g ∈ G. This representation is called the trivial representation, and it is often denoted simply by its representation module R. 2. Let G = Sn , the symmetric group on n symbols, V = R and ρ(g) = multiplication by ǫ(g), where ǫ(g) is the sign of g. This representation is called the sign representation of the symmetric group. 3. Let R = R, V = R2 and G = S3 . This group may be realized as the group of automorphisms of V generated by reflections in the three lines

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In terms of matrices this gives a representation   1 0 () 7→ 0 1   1 −1 (1, 2) 7→ 0 −1   0 1 (1, 3) 7→ 1 0   −1 0 (2, 3) 7→ −1 1   0 −1 (1, 2, 3) 7→ 1 −1   −1 1 (1, 3, 2) 7→ −1 0 where we have taken basis vectors in the direction of two of the lines of reflection, making an angle of 2π to each other. In fact these matrices define a representation of degree 2 3 over any ring R, because although the representation was initially constructed over R, in fact the matrices have integer entries which may be interpreted in every ring, and these matrices always multiply together to give a copy of S3 . 4. Let R = Fp , V = R2 and let G = Cp = hgi be cyclic of order p generated by an element g. One checks that the assignment r

ρ(g ) =



1 0 r 1



is a representation. In this case the fact that we have a representation is very much dependent on the choice of R as the field Fp : in characteristic 0 it would not work! We can think of representations in various ways, and one of them is that a representation is the specification of an action of a group on an R-module. Given a representation ρ : G → GL(V ), an element v ∈ V and a group element g ∈ G we get another module element ρ(g)(v). Sometimes we write just g · v or gv for this element. This rule for multiplication satisfies g · (λv + µw) = λg · v + µg · w (gh) · v = g · (h · v) 1·v =v for all g ∈ G, v, w ∈ V and λ, µ ∈ R. A rule for multiplication G × V → V satisfying these conditions is called a linear action of G on V . To specify a linear action of G on V is the same thing as specifying a representation of G on V , since given a representation we obtain a linear action as indicated above, and evidently given a linear action we may recover the representation.

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Another way to define a representation of a group is in terms of the group algebra. We define the group algebra RG (or R[G]) of G over R to be the free R-module with the elements of G as an R-basis, and with multiplication given on the basis elements by group multiplication. The elements of RG are the (formal) R-linear combinations of group elements, and the multiplication of the basis elements is extended to arbitrary elements P using bilinearity of the operation. A typical element of RG may be written g∈G ag g where ag ∈ R, and symbolically (

X

g∈G

ag g)(

X

h∈G

bh h) =

X X ( ag bh )k.

k∈G gh=k

More concretely, we may exemplify the definition by listing some elements of QS3 . The elements of S3 such as (1, 2) = 1 · (1, 2) are also elements of QS3 (they appear as basis elements), and () serves as the identity element of QS3 (as well as of S3 ). In general, elements of QS3 may look like (1, 2) − (2, 3) or 15 (1, 2, 3) + 6(1, 2) − 71 (2, 3). Here is a computation: ((1, 2, 3) + (1, 2))((1, 2) − (2, 3)) = (1, 3) + () − (1, 2) − (1, 2, 3). Having defined the group algebra, we may now define a representation of G over R to be a unital RG-module. The fact that this definition coincides with the previous ones is the content of the next proposition. (1.2) PROPOSITION. A representation of G over R has the structure of a unital RG-module; conversely, every unital RG-module provides a representation of G over R. Proof. Given a representation ρ : G → GL(V ) we define a module action of RG on V P P by ( ag g)v = ag ρ(g)(v). Given a RG-module V , the linear map ρ(g) : v 7→ gv is an automorphism of V and ρ(g1 )ρ(g2 ) = ρ(g1 g2 ) so ρ : G → GL(V ) is a representation. We have defined the group algebra without saying what an algebra is! For the record, an (associative) R-algebra is a ring A with a 1, equipped with a (unital) ring homomorphism R → A whose image lies in the center of A. The group algebra RG is indeed an example of an R-algebra. The group algebra gives another example of a representation, called the regular representation. In fact for any ring A we may regard A itself as a left A-module with the action of A on itself given by multiplication of the elements. We denote this left A-module by A A when we wish to emphasize the module structure, and this is the (left) regular representation of A. When A = RG we may describe action on RG RG by observing that each element g ∈ G acts on RG RG by permuting the basis elements in the fashion g · h = gh. Thus each g acts by a permutation matrix, namely a matrix in which in every row and column there is precisely one non-zero entry, and that non-zero entry is 1. The regular

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representation is an example of a permutation representation, namely one in which every group element acts by a permutation matrix. Regarding representations of G as RG-modules has the advantage that many definitions we wish to make may be borrowed from module theory. Thus we may study RG-submodules of an RG-module V , and if we wish we may call them subrepresentations of the representation afforded by V . To specify an RG-submodule of V it is necessary to specify an R-submodule W of V which is closed under the action of RG. This is equivalent to requiring that ρ(g)w ∈ W for all g ∈ G and w ∈ W . We say that a submodule W satisfying this condition is stable under G, or that it is an invariant submodule or invariant subspace (if R happens to be a field). Such an invariant submodule W gives rise to a homomorphism ρW : G → GL(W ) which is the subrepresentation afforded by W . (1.3) Examples. 1. Let C2 = {1, −1} be cyclic of order 2 and consider the representation ρ : C2 → GL(R2 )   1 0 1 7→ 0 1   1 0 −1 7→ 0 −1

One checks that the invariant subspaces are {0}, h 10 i, h 01 i, R2 and no others. Here

R2 = h 10 i ⊕ h 01 i is the direct sum of two invariant subspaces.
2. In Example 4 of 1.1 above, an elementary calculation shows that h 01 i is the only 1-dimensional invariant subspace, and so it is not possible to write the representation space V as the direct sum of two non-zero invariant subspaces. We also have the notions of a homomorphism and an isomorphism of RG-modules. Since RG has as a basis the elements of G, to check that an R-linear homomorphism f : V → W is in fact a homomorphism of RG modules, it suffices to check that f (gv) = gf (v) for all g ∈ G — we do not need to check for every x ∈ RG. By means of the identification of RG-modules with representations of G (in the first definition given here) we may refer to homomorphisms and isomorphisms of group representations. In many books the algebraic condition on the representations which these notions entail is written out explicitly, and two representations which are isomorphic are also said to be equivalent. If V and W are RG-modules then we may form their (external) direct sum V ⊕ W , which is the same as the direct sum of V and W as R-modules together with an action of G given by g(v, w) = (gv, gw). We also have the notion of the internal direct sum of RG-modules and write U = V ⊕ W to mean that U has RG-submodules V and W satisfying U = V + W and V ∩ W = 0. In this situation we also say that V and W are direct summands of U . We already met this property in (1.3) above, whose first part is an example of a representation which is a direct sum of two non-zero subspaces; however, the second part of (1.3) provides an example of a subrepresentation which is not a direct summand.

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We come now to our first non-trivial result, and one which is fundamental to the study of representations over fields of characteristic zero. Here, as in many other places, we require that the ring R be a field, and in this situation we use the symbols R or k instead of R. (1.4) THEOREM (Maschke). Let V be a representation of the finite group G over a field F in which |G| is invertible. Let W be an invariant subspace of V . Then there exists an invariant subspace W1 of V such that V = W ⊕ W1 as representations. Proof. Let π : V → W be any projection of V onto W as vector spaces, i.e. a linear transformation such that π(w) = w for all w ∈ W . Since F is a field, we may always find such a projection by finding a vector space complement to W in V , and projecting off the complementary factor. Then V = W ⊕ Ker(π) as vector spaces, but Ker(π) is not necessarily invariant under G. Consider the map π′ =

1 X gπg −1 : V → V. |G| g∈G

Then π ′ is linear and if w ∈ W then π ′ (w) =

1 X gπ(g −1w) |G| g∈G

=

1 X −1 gg w |G| g∈G

1 |G|w |G| = w. =

Since π ′ (v) ∈ W for all v ∈ V , π ′ is a projection onto W and so V = W ⊕ Ker(π ′ ). We show finally that Ker(π ′ ) is an invariant subspace. If h ∈ G and v ∈ Ker(π ′ ) then π ′ (hv) =

1 X gπ(g −1hv) |G| g∈G

1 X h(h−1 g)π((h−1 g)−1 v) = |G| g∈G

′

= hπ (v) =0 since as g ranges over the elements of G, so does h−1 g. This shows that hv ∈ Ker(π ′ ) and so Ker(π ′ ) is an invariant subspace.

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Let A be a ring with a 1. An A-module V is said to be simple or irreducible if V has no A-submodules other than 0 and V . A module which is the direct sum of simple submodules is said to be semisimple or completely reducible and we saw in (1.3) examples of modules, one of which was semisimple and the other of which was not. We know from the Jordan-H¨older theorem in module theory that in some sense simple modules are the building blocks for arbitrary modules of finite composition length. One way in which these building blocks may be combined is as a direct sum — a construction we feel we understand quite well — giving a semisimple module, but there may be other modules which are not constructed from simple modules in this way. The next results relate the property of semisimplicity to the property which appears in the statement of Maschke’s theorem, namely that every submodule of a module is a direct summand. Our immediate application of this will be an interpretation of Maschke’s theorem, but the results have application in greater generality in situations where R is not a field, or when |G| is not invertible in R. To simplify the exposition we have imposed a finiteness condition in the statement of each result, thereby avoiding arguments which use Zorn’s lemma. These finiteness conditions can be removed, and we leave the details to the reader in an exercise at the end of this section. In the special case when the ring A is a field and A-modules are vector spaces the next result is a familiar statement from linear algebra. (1.5) LEMMA. Let A be a ring with a 1 and suppose that U = S1 + · · · + Sn is an A-module which can be written as the sum of finitely many simple modules S1 , . . . , Sn . If V is any submodule of U there is a subset I = {i1 , . . . , ir } of {1, . . . , n} such that U = V ⊕ Si1 ⊕ · · · Sir . In particular, (1) V is a direct summand of U , and (2) (taking V = 0), U is the direct sum of some subset of the Si , and hence is necessarily semisimple. Proof. Choose a subset I maximal subject to the condition that the sum W = L V ⊕ i∈I Si is a direct sum. Note that I = ∅ has this property, so we are indeed taking a maximal element of a non-empty set. We show that W = U . If W 6= U then Sj 6⊆ W for some j. Now Sj ∩ W = 0, being a proper submodule of Sj , so Sj + W = Sj ⊕ W and we obtain a contradiction to the maximality of I. Therefore W = U . (1.6) PROPOSITION. Let A be a ring with a 1 and let U be an A-module. The following are equivalent. (1) U can be expressed as a direct sum of finitely many simple A-submodules. (2) U can be expressed as a sum of finitely many simple A-submodules. (3) U has finite composition length and has the property that every submodule of U is a direct summand of U .

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When these three conditions hold, every submodule of U and every factor module of U may also be expressed as the direct sum of finitely many simple modules. Proof. The implication (1) ⇒ (2) is immediate and the implications (2) ⇒ (1) and (2) ⇒ (3) follow from Lemma 1.5. To show that (3) ⇒ (1) we argue by induction on the composition length of U , and first observe that hypothesis (3) passes to submodules of U . For if V is a submodule of U and W is a submodule of V then U = W ⊕ X for some submodule X, and now V = W ⊕ (X ∩ V ) by the modular law (Exercise 2). Proceeding with the induction argument, when U has length 1 it is a simple module, and so the induction starts. If U has length greater than 1, it has a submodule V and by condition (3), U = V ⊕ W for some submodule W . Now both V and W inherit condition (3) and are of shorter length, so by induction they are direct sums of simple modules and hence so is U . We have already observed that every submodule of U inherits condition (3), and so satisfies condition (1) also. Every factor module of U has the form U/V for some submodule V of U . If condition (3) holds than U = V ⊕ W for some submodule W which we have just observed satisfies condition (1), and hence so does U/V since U/V ∼ = W. We now present a different statement of Maschke’s theorem. The statement remains correct if the words ‘finite-dimensional’ are removed from it, but we leave the proof of this stronger statement to the exercises. (1.7) COROLLARY. Let R be a field in which |G| is invertible. Then every finitedimensional RG-module is semisimple. Proof. This combines Theorem 1.4 with the equivalence of the statements of Proposition 1.6. This theorem puts us in very good shape if we want to know about the representations of a finite group over a field in which |G| is invertible — for example any field of characteristic zero. To obtain a description of all possible finite-dimensional representations we need only describe the simple ones, and then arbitrary ones are direct sums of these. The following corollaries to Lemma 1.5 will not immediately be used, but we present them here because they have the same flavor as the results just considered. They could be omitted at this point, and read as they are needed. (1.8) COROLLARY. Let A be a ring with a 1, and let U be an A-module of finite composition length. (1) The sum of all the simple submodules of U is a semisimple module, which is the unique largest semisimple submodule of U .

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(2) The sum of all submodules of U isomorphic to some given simple module S is a submodule isomorphic to a direct sum of copies of S. It is the unique largest submodule of U with this property. Proof. The submodules described can be expressed as the sum of finitely many submodules by the finiteness condition on U . They are the unique largest submodules with their respective properties since they contain all simple submodules (in case (1)), and all submodules isomorphic to S (in case (2)). The largest semisimple submodule of a module U is called the socle of U , and is denoted Soc(U ). (1.9) COROLLARY. Let U = S1a1 ⊕ · · · ⊕ Srar be a semisimple module over a ring A with a 1, where the Si are non-isomorphic simple modules and the ai are the multiplicities to which they occur as summands of U . Then each submodule Siai is uniquely determined and is characterized as the unique largest submodule of U expressible as a direct sum of copies of Si . Proof. It suffices to show that Siai contains every submodule of U isomorphic to Si . If T is any non-zero submodule of U not contained in Siai then for some j 6= i its projection to a summand Sj must be non-zero. If we assume that T is simple this projection will be an isomorphism T ∼ = Sj . Thus all simple submodules isomorphic to Si are contained in ai the summand Si . Exercises for Section 1. 1. In Example 1 of 1.3 prove that there are no more invariant subspaces other than the ones listed. 2. (The modular law.) Let A be a ring and U = V ⊕ W an A-module which is the direct sum of A-modules V and W . Show by example that if X is any submodule of U then it need not be the case that X = (V ∩ X) ⊕ (W ∩ X). Show that if we make the assumption that V ⊆ X then it is true that X = (V ∩ X) ⊕ (W ∩ X). 3. Suppose that ρ is a finite-dimensional representation of a finite group G over C. Show that for each g ∈ G the matrix ρ(g) is diagonalizable. 4. Let ρ1 : G → GL(V ) ρ2 : G → GL(V ) be two representations of G on the same R-module V which are injective as homomorphisms. (One says that such a representation is faithful.) Consider the three statements (a) the RG-modules given by ρ1 and ρ2 are isomorphic, (b) the subgroups ρ1 (G) and ρ2 (G) are conjugate in GL(V ),

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(c) for some automorphism α ∈ Aut(G) the representations ρ1 and ρ2 α are isomorphic. Show that (a) ⇒ (b) and that (b) ⇒ (c). Show also that if α ∈ Aut(G) is an inner automorphism (i.e. one of the form ‘conjugation by g’ for some g ∈ G) then ρ1 and ρ1 α are isomorphic. 5. One form of the Jordan-Zassenhaus theorem states that for each n, GL(n, Z) (that is, Aut(Zn )) has only finitely many conjugacy classes of subgroups of finite order. Assuming this, show that for each finite group G and each integer n there are only finitely many isomorphism classes of representations of G on Zn . 6. (a) Write out a proof of Maschke’s theorem in the case of representations over C along the following lines. Given a representation ρ : G → GL(V ) where V is a vectorspace over C, let ( , ) be any positive definite Hermitian form on V . Define a new form ( , )1 on V by (v, w)1 =

1 X (gv, gw). |G| g∈G

Show that ( , )1 is a positive definite Hermitian form, preserved under the action of G, i.e. (v, w)1 = (gv, gw)1 always. If W is a subrepresentation of V , show that V = W ⊕ W ⊥ as representations. (b) Show that any finite subgroup of GL(n, C) is conjugate to a subgroup of U (n, C) (the unitary group, consisting of n × n complex matrices A satisfying AA¯T = I). Show that any finite subgroup of GL(n, R) is conjugate to a subgroup of O(n, R) (the orthogonal group consisting of n × n real matrices A satisfying AAT = I). 7. Let U = S1 ⊕ · · · ⊕ Sr be an A-module which is the direct sum of finitely many simple modules S1 , . . . , Sr . Show that if T is any simple submodule of U then T ∼ = Si for some i. 8. Suppose U is an A-module for which we have two expressions U∼ = S1a1 ⊕ · · · ⊕ Srar ∼ = S1b1 ⊕ · · · ⊕ Srbr where S1 , . . . , Sr are non-isomorphic simple modules. Show that ai = bi for all i. 2 2 9. Let G = hx, y x = y = 1 = [x, y]i be the Klein four-group, R = F2 , and consider the two representations ρ1 and ρ2 specified on the generators of G by     1 1 0 1 0 1 ρ1 (x) =  0 1 0  , ρ1 (y) =  0 1 0  0 0 1 0 0 1 and



1 ρ2 (x) =  0 0

 0 0 1 1, 0 1



1 ρ2 (y) =  0 0

 0 1 1 0. 0 1

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Calculate the socles of these two representations. 10. Let G = Cp = hxi and R = Fp for some prime p ≥ 3. Consider the two representations ρ1 and ρ2 specified by 

1  ρ1 (x) = 0 0

 1 0 1 1 0 1



1  and ρ2 (x) = 0 0

 1 1 1 0. 0 1

Calculate the socles of these two representations. Show that the second representation is the direct sum of two non-zero subrepresentations. 11. (a) Using 1.6 show that if A is a ring for which the regular representation A A is semisimple, then every finitely generated A-module is semisimple. (b) Extend the result of part (a), using Zorn’s lemma, to show that if A is a ring for which the regular representation A A is semisimple, then every A-module is semisimple. 12. Let U be a module for a ring A with a 1. Show that the following three statements are equivalent. (1) U is a direct sum of simple A-submodules. (2) U is a sum of simple A-submodules. (3) every submodule of U is a direct summand of U . [Use Zorn’s lemma to prove a version of Lemma 1.5 which has no finiteness hypothesis and then copy Proposition 1.6. This deals with all implications except (3) ⇒ (2). For that, use the fact that A has a 1 and hence every (left) ideal is contained in a maximal (left) ideal, combined with condition (3), to show that every submodule of U has a simple submodule. Consider the sum of all simple submodules of U and show that it equals U .] 13. Let RG be the group algebra of a finite group G over a commutative ring R with 1. Let S be a simple RG-module and let I be the anihilator in R of S, that is I = {r ∈ R rx = 0 for all x ∈ S}.

Show that I is a maximal ideal in R. [This question requires some familiarity with standard commutative algebra. We conclude from this result that when considering simple RG modules we may reasonably assume that R is a field, since S may naturally be regarded as an (R/I)G-module and R/I is a field.]

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2. The structure of algebras for which every module is semisimple. In this section we work with an abstract finite-dimensional algebra A over a field k (unless greater generality is stated). In studying the simple A-modules it is no loss of generality to assume that we are working over a field, as explained in an exercise at the end of the last section. Furthermore, every simple A-module is isomorphic to one of the form A/I where I is some maximal left ideal of A. This is because if x ∈ S is any non-zero element of a simple module S then the mapping of left modules A → S specified by a 7→ ax is surjective, by simplicity of S, and so S ∼ = A/I where I is the kernel, which is a maximal ideal again by simplicity of S. The following general result is basic, and will be used time and time again. (2.1) THEOREM (Schur’s Lemma). Let S1 and S2 be simple A-modules where A is a ring with 1. Then HomA (S1 , S2 ) = 0 unless S1 ∼ = S2 , in which case EndA (S1 ) is a division ring. In case A is a finite-dimensional algebra over an algebraically closed ground field k, then every endomorphism of S1 is scalar multiplication by an element of k. Thus EndA (S1 ) ∼ = k. Proof. Suppose θ : S1 → S2 is a non-zero morphism. Then 0 6= θ(S1 ) ⊆ S2 , so θ(S1 ) = S2 by simplicity of S2 and θ is surjective. Thus Ker θ 6= S1 , so Ker θ = 0 by simplicity of S1 , and θ is injective. Therefore θ is invertible, S1 ∼ = S2 and EndA (S1 ) is a division ring. If k is algebraically closed, let λ be an eigenvalue of θ. Now (θ − λI) : S1 → S1 is a singular endomorphism of A-modules, so θ − λI = 0 and θ = λI. The next result is straightforward and seemingly innocuous, but it has an important consequence for representation theory. It is the main tool in recovering the structure of an algebra from its representations. We use the notation Aop to denote the opposite ring of A, namely the ring which has the same set and the same addition as A, but in which there is a new multiplication · given by a · b = ba. (2.2) LEMMA. For any ring A with a 1, EndA (A A) ∼ = Aop . Proof. The inverse isomorphisms are φ 7→ φ(1) (a 7→ ax) ← x. There are several things here which need to be checked: that the second assignment does take values in EndA (A A), that the morphisms are ring homomorphisms, and that they are mutually inverse. We leave most of this to the reader, observing only that under the first homomorphism a composite θφ is sent to (θφ)(1) = θ(φ(1)) = θ(φ(1)1) = φ(1)θ(1), so that this is indeed a homomorphism to Aop .

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Observe that the proof of Lemma 2.2 establishes that every endomorphism of the regular representation is of the form ‘right multiplication by some element’. We say that a ring A with 1 is semisimple if the regular representation A A is semisimple. There are other equivalent definitions of a semisimple ring which will be explored further in Chapter 6. A ring A with 1 all of whose modules are semisimple is itself called semisimple. By Exercise 1.11 it is equivalent to suppose that the regular representation A A is semisimple. It is also equivalent to suppose that the Jacobson radical of the ring is zero, but we will not deal with this point of view until Chapter 6. (2.3) THEOREM (Artin-Wedderburn). Let A be a finite-dimensional algebra over a field with the property that every finite-dimensional module is semisimple. Then A is a direct sum of matrix algebras over division rings. Specifically, if AA

∼ = S1n1 ⊕ · · · ⊕ Srnr

where the S1 , . . . , Sr are non-isomorphic simple modules occuring with multiplicities ni in the regular representation, then A∼ = Mn1 (D1 ) ⊕ · · · ⊕ Mnr (Dr ) where Di = EndA (Si )op . More is true: every such direct sum of matrix algebras is a semisimple algebra, and each matrix algebra over a division ring is a simple algebra, namely one which has no 2-sided ideals apart from the zero ideal and the whole ring (see the exercises). Furthermore, the matrix algebra summands are uniquely determined as subsets of A (although the module decomposition of A A is usually only determined up to isomorphism). The uniqueness of the summands will be established in Proposition 3.22. Notice that if k happens to be algebraically closed then by Schur’s lemma the division rings Di which appear must all be equal to k. Proof. We first observe that if we have a direct sum decomposition U = U1 ⊕ · · · ⊕ Ur of a module U then EndA (U ) is isomorphic to the algebra of r × r matrices in which the i, j entries lie in HomA (Uj , Ui ). This is because any endomorphism φ : U → U may be writen as components φ = (φij ) where φij : Uj → Ui , and in terms of components these n endomorphisms compose in the manner of matrix multiplication. Since HomA (Sj j , Sini ) = 0 if i 6= j by Schur’s lemma, the decomposition of A A gives n EndA (A A) ∼ = EndA (S1 1 ) ⊕ · · · ⊕ EndA (Srnr ) op and furthermore EndA (Sini ) ∼ = Mni (Di ). By Lemma 2.2 we identify EndA (A A) as Aop .

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(2.4) COROLLARY. Let A be a finite-dimensional semisimple algebra over a field k which is algebraically closed. In any decomposition AA

= S1n1 ⊕ · · · ⊕ Srnr

where the Si are pairwise non-isomorphic simple modules we have that S1 , . . . , Sr is a complete set of representatives of the isomorphism classes of simple A-modules, ni = dimk Si and dimk A = n21 + · · · + n2r . Proof. All isomorphism types of simple modules must appear in the decomposition because every simple module can be expressed as a homomorphic image of A A (as observed at the start of this section), and so must be a homomorphic image of one of the modules Si . Since k is algebraically closed all the division rings Di coincide with k by Schur’s lemma, and EndA (Sini ) ∼ = Mni (k). The ring decomposition A = Mn1 (k)⊕· · ·⊕Mnr (k) immediately gives dimk A = n21 + · · · + n2r . From the way this decomposition was constructed as an endomorphism ring in 2.3 we see that Mni (k) has non-zero action on the summand Sini , n and zero action on the other summands Sj j with j 6= i. It follows that as left A-modules, Mni (k) ∼ = Sini since both sides are isomorphic to the quotient of A by the elements which Mni (k) annihilates. Hence dimk Mni (k) = n2i = dimk Sini = ni dim Si , and so dim Si = ni . Let us now restate what we have proved specifically in the context of group representations. (2.5) COROLLARY. Let G be a finite group and k a field in which |G| is invertible. (1) As a ring, kG is a direct sum of matrix algebras over division rings, (2) Suppose in addition that k is algebraically closed, let S1 , . . . , Sr be a set of representatives of the simple kG-modules (up to isomorphism) and put di = dimk Si . Then di equals the multiplicity with which Si is a summand of the regular representation of G, and |G| = d21 + · · · + d2r . Part (2) of this result provides a numerical criterion which enables us to say when we have constructed all the simple modules of a group over an algebraically closed field k in which |G| is invertible. For if we have constructed a set of non-isomorphic simple modules with the property that the squares of their degrees sum to |G|, then we have a complete set. While this is an easy condition to verify, it will be superceded later on by the even more straightforward criterion that the number of simple kG-modules (with the same hypotheses on k) equals the number of conjugacy classes of elements of G. Once we have P 2 proved this, the formula di = |G| allows the degree of the last simple representation to be determined once the others are known.

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(2.6) Example. Over R we have constructed for S3 the trivial representation, the sign representation which is also of dimension 1 but not the same as the trivial representation, and a 2-dimensional representation which is simple because visibly no 1-dimensional subspace of the plane is invariant under the group action. Since 12 + 12 + 22 = |S3 | we have constructed all the simple representations. Exercises for Section 2. 1. Let A be a finite-dimensional semisimple algebra. Show that A has only finitely many isomorphism types of modules in each dimension. [This is not in general true for algebras which are not semisimple: The representations of C2 ×C2 = hx, yi over F2 specified for each λ ∈ F2 by     1 0 0 0 1 0 0 0 0 1 λ 0 1 1 0 0 ρλ (y) =  ρλ (x) =    0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 all have dimension 4 and are non-isomorphic.] 2. Using Exercises 1.5 and 2.1 (which you may assume without proof), show that if k is any field of characteristic 0 then for each natural number m, GLn (k) has only finitely many conjugacy classes of subgroups of order m. [In view of the comment to problem 1, the same is not true when when k = F2 .] 3. Let D be a division ring and n a natural number. (a) Show that the natural Mn (D)-module consisting of column vectors of length n is a simple module. (b) Show that Mn (D) is semisimple and has up to isomorphism only one simple module. (c) Show that every algebra of the form Mn1 (D1 ) ⊕ · · · ⊕ Mnr (Dr ) is semisimple. (d) Show that Mn (D) is a simple ring, namely one in which the only 2-sided ideals are the zero ideal and the whole ring. 4. Prove the following extension of 2.4: THEOREM. Let A be a finite-dimensional semisimple algebra, S a simple A-module and D = EndA (S). Then S may be regarded as a module over D and the multiplicity of S as a summand of A A equals dimD S. 5. Using the fact that Mn (k) has a unique simple module prove the Noether-Skolem theorem, that any algebra automorphism of Mn (k) is inner, i.e. of the form conjugation by some invertible matrix.

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6. Show that for any field k we have Mn (k) ∼ = Mn (k)op , and in general for any division ring D that given any positive integer n, Mn (D) ∼ = Mn (D)op if and only if D ∼ = Dop . 7. Let A be any algebra. An element e ∈ A is called idempotent if and only if e2 = e. (a) Let V be any A-module. Show that an endomorphism e : V → V is a projection onto a subspace W if and only if e is idempotent as an element of EndA (V ). (b) Show that direct sum decompositions V = W1 ⊕ W2 as A-modules are in bijection with expressions 1 = e + f in EndA (V ), where e and f are idempotent elements with ef = f e = 0. (In case ef = f e = 0, e and f are called orthogonal.) (c) Let e1 , e2 ∈ EndA (V ) be idempotent elements. Show that e1 (V ) ∼ = e2 (V ) as Amodules if and only if e1 and e2 are conjugate under the unit group EndA (V )∗ (i.e. there exists an A-endomorphism α : V → V such that e2 = αe1 α−1 ). (d) An idempotent element e is called primitive if it cannot be expressed as a sum of orthogonal idempotent elements in a non-trivial way. Show that e ∈ EndA (V ) is primitive if and only if e(V ) has no (non-trivial) direct sum decomposition. (In this case V is said to be indecomposable.) (e) Show that all primitive idempotent elements in Mn (k) are conjugate under the action of the unit group GLn (k). Write down explicitly any primitive idempotent element in M3 (k).

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3. Characters The characters of a finite group take values in a field of characteristic zero which we may always take to be a subfield of the complex numbers, and so frequently in this section we will restrict our attention to the complex numbers C. When ρ : G → GL(V ) is a finite-dimensional representation of G over C we define the character χ of ρ to be the function χ:G→C given by χ(g) = tr(ρ(g)), the trace of the linear map ρ(g). We say that the representation ρ and the representation space V afford the character χ, and we may write χρ or χV when we wish to specify this character more precisely.

(1) (2) (3) (4)

(3.1) PROPOSITION. χ(1) is the degree of ρ. For every g ∈ G we have χ(g −1 ) = χ(g), the complex conjugate. χ(hgh−1 ) = χ(g) for all g, h ∈ G. If V and W are isomorphic CG-modules then χV (g) = χW (g) for all g ∈ G.

Proof. (1) is immediate because the identity of the group must act as the identity matrix. (2) Each element g has finite order so its eigenvalues λ1 , . . . , λn are roots of unity. The −1 −1 ) = λ1 +· · ·+λn = χ(g). inverse g −1 has eigenvalues λ−1 1 , . . . , λn = λ1 , . . . , λn and so χ(g (3) results from the fact that tr(ab) = tr(ba) for endomorphisms a and b, so that χ(hgh−1 ) = tr ρ(hgh−1 ) = tr(ρ(h)ρ(g)ρ(h−1)) = tr ρ(g) = χ(g). (4) Suppose that ρV and ρW are the representations of G on V and W , and that we have an isomorphism of CG-modules α : V → W . Then αρV (g) = ρW (g)α for all g ∈ G, so that χW (g) = tr ρW (g) = tr(αρV (g)α−1 ) = tr ρV (g) = χV (g). Part (4) of the above result is a great convenience since when talking about characters we do not have to worry about the possibility that two modules may be isomorphic but not actually the same. From part (3) of 3.1 we see that the character of a representation takes the same value on all elements in a conjugacy class of G. The table of complex numbers whose rows are indexed by the isomorphism types of simple representations of G, whose columns are indexed by the conjugacy classes of G and whose entries are the values of the characters of the simple representations on representatives of the conjugacy classes is called the character table of G. It is usual to index the first column of a character table by the (conjugacy class of the) identity, and to put the character of the trivial representation as the top row. With this convention the top row of every character table will be a row of 1’s, and the first column will list the degrees of the simple representations. At the top of the table it is usual to list two rows, the second of which (immediately above a horizontal line) is a list

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of representatives of the conjugacy classes of elements of G, in some notation. The very top row lists the value of |CG (g)| for the element g underneath. (3.2) Example. We present the character table of S3 . We saw at the end of Section 2 that we already have a complete list of the simple modules for S3 , and the values of their characters on representatives of the conjugacy classes of S3 are computed from the matrices which give these representations. Centralizer orders Conjugacy class representative Trivial Sign

6 2 3 1 (12) (123) 1 1 1 1 −1 1 2 0 −1

TABLE: The character table of S3 . We will see that the character table has remarkable properties, among which are that it is always square, and its rows (and also its columns) satisfy certain orthogonality relations. Our next main goal is to state and prove these results. In order to do this we first introduce three ways to construct new representations of a group from existing ones. These constructions have validity no matter what ring R we work over, although in the application to the character table we will suppose that R = C. Suppose that V and W are representations of G over R. The R-module V ⊗R W acquires an action of G by means of the formula g · (v ⊗ w) = gv ⊗ gw, thereby making the tensor product into a representation. This is what is called the tensor product of the representations V and W , but it is not the only occurrence of tensor products in representation theory, and as the other ones are different this one is sometimes also called the Kronecker product. The action of G on the Kronecker product is called the diagonal action. For the second construction we form the R-module HomR (V, W ). This acquires an action of G by means of the formula (g·f )(v) = gf (g −1v) for each R-linear map f : V → W and g ∈ G. The third construction is the particular case of the second in which we take W to be the trivial module R. We write V ∗ = HomR (V, R) and the action is (g · f )(v) = f (g −1 v). This representation is called the dual or contragredient representation of V , and it is usually only considered when V is free as an R-module. If R happens to be a field and we have bases v1 , . . . , vm for V and w1 , . . . , wn for W then V ⊗ W has a basis {vi ⊗ wj 1 ≤ i ≤ m, 1 ≤ j ≤ n} and V ∗ has a dual basis vˆ1 , . . . , vˆm . With respect to these bases an element g ∈ G acts on V ⊗ W with the matrix which is the tensor product of the two matrices giving its action on V and W , and on V ∗ it acts with the transpose of the inverse of the matrix in its action on V . The tensor product of two matrices is not seen so often these days. If (apq ), (brs ) are an m × m matrix and an

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n × n matrix their tensor product is the mn × mn matrix (cij ) where if i = (p − 1)n + r and j = (q − 1)n + s with 1 ≤ p, q ≤ m and q ≤ r, s ≤ n then cij = apq brs . For example, 

a b c d



⊗



e g

f h





ae  ag = ce cg

af ah cf ch

be bg de dg

 bf bh  . df dh

If α : V → V and β : W → W are endomorphisms, then the matrix of α⊗β : V ⊗W → V ⊗W is the tensor product of the matrices which represent α and β. We see from this that tr(α ⊗ β) = tr(α) tr(β). In the following result we consider the sum and product of characters, which are defined pointwise by the formulas (χV + χW )(g) = χV (g) + χW (g) (χV · χW )(g) = χV (g) · χW (g). (3.3) PROPOSITION. Let V and W be finite-dimensional representations of G over a field k of characteristic zero. (1) V ⊕ W has character χV + χW . (2) V ⊗ W has character χV · χW . (3) V ∗ has character χV ∗ (g) = χV (g −1 ) = χV (g). (4) Homk (V, W ) ∼ = V ∗ ⊗k W as kG-modules, and this representation has character equal to χV ∗ · χW . We will see in the proof that the isomorphism of modules in part (4) is in fact valid without restriction on the characteristic of the field k. Proof. (1), (2) and (3) are immediate on taking matrices for the representations. As for (4), we define a linear map α : V ∗ ⊗ W → Homk (V, W ) f ⊗ w 7→ (v 7→ f (v) · w), this being the specification on basic tensors. We show that α is injective. Assuming that Pn w1 , . . . , wn is a basis of W , every element of V ∗ ⊗ W can be written i=1 fi ⊗ wi where P fi : V → k. Such an element is sent by α to the map v 7→ fi (v)wi , where v ∈ V . If P this is the zero mapping then fi (v)wi = 0 for all v ∈ V , and by independence of the wi Pn we have fi (v) = 0 for all v ∈ V . Thus i=1 fi ⊗ wi = 0 which shows that α is injective.

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Since V ∗ ⊗ W and Homk (V, W ) have the same dimension (equal to dim V · dim W ), α is a vector space isomorphism. It is also a map of kG-modules, since for g ∈ G, α(g(f ⊗ w)) = α(gf ⊗ gw) = (v 7→ (gf )(v) · gw) = (v 7→ g(f (g −1v)w)) = g(v 7→ f (v)w) = gα(f ⊗ w).

A fundamental notion in dealing with group actions is that of fixed points. If V is an RG-module we define the fixed points V G = {v ∈ V gv = v for all g ∈ G}. This is the largest RG-submodule of V on which G has trivial action. (3.4) LEMMA. Over any ring R, HomR (V, W )G = HomRG (V, W ). Proof. An R-linear map f : V → W is a morphism of RG-modules precisely if it commutes with the action of G, which is to say f (gv) = gf (v) for all g ∈ G and v ∈ V , or in other words gf (g −1v) = f (v) always. This is exactly the condition that f is fixed under the action of G. The next result is an abstraction of the idea which was used in proving Maschke’s theorem, where the application was to the RG-module HomR (V, V ). We will use this idea a second time in proving the orthogonality relations for characters. (3.5) LEMMA. Let V be an RG-module where R is a ring in which |G| is invertible. Then 1 X g :V →VG |G| g∈G

is a map of RG-modules which is projection onto the fixed points of V . In particular, V G is a direct summand of V as an RG-module. When R is a field of characteristic zero we have 1 X g) = dim V G . tr( |G| g∈G

P 1 Proof. Let π : V → V denote the map ‘multiplication by |G| g∈G g’. We check that π is a linear map, and it commutes with the action of G since for h ∈ G and v ∈ V we

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have π(hv) = (

20

1 X gh)v |G| g∈G

= π(v) 1 X =( hg)v |G| g∈G

= hπ(v) since as g ranges through the elements of G so do gh and hg. The same equations show that every vector of the form π(v) is fixed by G. Furthermore, if v ∈ V G then π(v) =

1 X 1 X gv = v=v |G| |G| g∈G

g∈G

so π is indeed projection onto V G . There is one more ingredient we describe before stating the orthogonality relations for characters. We define an inner product on characters, but this does not make sense without some further explanation, because an inner product must be defined on a vector space and characters do not form a vector space. They are, however, elements in a vector space, namely the vector space of class functions on G. A class function on G is a function G → C which is constant on each conjugacy class of G. Such functions are in bijection with the functions from the set of conjugacy classes of G to C, a set of functions which we may denote Ccc(G) where cc(G) is the set of conjugacy classes of G. These functions become an algebra when we define addition, multiplication and scalar multiplication pointwise on the values of the function. In other words, (χ · ψ)(g) = χ(g)ψ(g), (χ + ψ)(g) = χ(g) + ψ(g) and (λχ)(g) = λχ(g) where χ, ψ are class functions and λ ∈ C. If G has n conjugacy classes, this algebra is isomorphic to Cn , the direct sum of n copies of C, and is semisimple. We define a Hermitian bilinear form on the complex vector space of class functions on G by means of the formula 1 X hχ, ψi = χ(g)ψ(g). |G| g∈G

As well as the usual identities which express bilinearity and the fact that the form is evidently Hermitian, it satisfies hχφ, ψi = hχ, φ∗ ψi where φ∗ (g) = φ(g) is the class function obtained by complex conjugation. If χ and ψ happen to be characters of a representation we have χ(g) = χ(g −1 ), ψ ∗ is the character of

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the contragredient representation, and we obtain further expressions for the bilinear form: hχ, ψi =

1 X χ(g −1 )ψ(g) |G| g∈G

=

1 X χ(g)ψ(g −1) |G| g∈G

=

1 X χ(g)ψ(g), |G| g∈G

where the second equality is obtained by observing that as g ranges over the elements of G, so does g −1 . With all this preparation we can now state and prove the orthogonality relations for the rows of the character table. The picture will be completed once we have shown that the character table is square and deduced the orthogonality relations for columns in 3.16 and 3.17. (3.6) THEOREM (Row Orthogonality relations). (1) If χ is the character of a simple representation over C then hχ, χi = 1. (2) If χ and ψ are the characters of non-isomorphic simple representations over C then hχ, ψi = 0. Proof. Suppose that V and W are simple complex representations affording characters χ and ψ. By 3.3 the character of HomC (V, W ) is χ · ψ. By 3.4 and 3.5 dim HomCG (V, W ) = tr(

1 X g) in its action on HomC (V, W ) |G| g∈G

1 X = χ(g)ψ(g) |G| g∈G

= hχ, ψi. Schur’s lemma asserts that this number is 1 if V ∼ = W , and 0 if V 6∼ = W. We will describe many consequences of the orthogonality relations, and the first is that they provide a way of determining the decomposition of a given representation as a direct sum of simple representations. This procedure is similar to the way of finding the coefficients in the Fourier expansion of a function using orthogonality of the functions sin(mx) and cos(nx).

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(3.7) COROLLARY. Let V be a CG-module. In any expression V = S1n1 ⊕ · · · ⊕ Srnr in which S1 , . . . , Sr are non-isomorphic simple modules, we have ni = hχV , χi i where χV is the character of V and χi is the character of Si . In particular ni is determined by V independently of the choice of decomposition. (3.8) Example. Let G = S3 and denote by C the trivial representation, ǫ the sign representation and V the 2-dimensional simple representation over C. We decompose the 4-dimensional representation V ⊗ V as a direct sum of simple representations. Since the values of the character χV give the row of the character table 0

− 1,

χV ⊗V : 4

0 1.

χV : 2 V ⊗ V has character values

Thus 1 (4 · 1 + 0 + 2 · 1 · 1) = 1 6 1 hχV ⊗V , χǫ i = (4 · 1 + 0 + 2 · 1 · 1) = 1 6 1 hχV ⊗V , χV i = (4 · 2 + 0 − 2 · 1 · 1) = 1 6 hχV ⊗V , χC i =

and we deduce that V ⊗V ∼ = C ⊕ ǫ ⊕ V. (3.9) COROLLARY. For finite-dimensional complex representations V and W we have V ∼ = W if and only if χV = χW . Proof. We saw in 3.1 that if V and W are isomorphic then they have the same character. Conversely, if they have the same character they both may be decomposed as a direct sum of simple representations by 1.7, and by 3.7 the multiplicities of the simples in these two decompositions must be the same. Hence the representations are isomorphic.

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The next result is a criterion for a representation to be simple. An important step in studying the representation theory of a group is to construct its character table, and one proceeds by compiling a list of the simple characters which at the end of the calculation will be complete. At any stage one has a partial list of simple characters, and considers some (potentially) new character. One finds the multiplicity of each previously obtained simple character as a summand of the new character, and subtracts off the these simple characters to the correct multiplicity. What is left is a character all of whose simple components are new simple characters. This character will itself be simple precisely if the following easily verified criterion is satisfied. (3.10) COROLLARY. If χ is the character of a complex representation V then hχ, χi is a positive integer, and equals 1 if and only if V is simple. ∼ S n1 ⊕ · · · ⊕ S nr and then hχ, χi = Pr n2 is a positive Proof. We may write V = r 1 i=1 i integer, which equals 1 precisely if one ni is 1 and the others are 0. (3.11) Example. We construct the character table of S4 , since it illustrates some techniques in finding simple characters. 24 4 8 4 3 1 (12) (12)(34) (1234) (123) 1 1 1 1 1 1 −1 1 −1 1 2 0 2 0 −1 3 −1 −1 1 0 3 1 −1 −1 0 TABLE: The character table of S4 . Immediately above the horizontal line we list representatives of the conjugacy classes of elements of S4 , and above them the orders of their centralizers. The first row below the line is the character of the trivial representation, and below that is the character of the sign representation. There is a homomorphism σ : S4 → S3 specified by (12) 7→ (12), (34) 7→ (12), (23) 7→ (23), (123) 7→ (123) which has kernel the normal subgroup h(12)(34), (13)(24)i. (One way to obtain this homomorphism is to identify S4 as the group of rotations of a cube and observe that each rotation gives rise to a permutation of the three pairs of opposite faces.) Any representation ρ : S3 → GL(V ) gives rise to a representation ρσ of S4 obtained by composition with σ, and if we start with a simple representation of S3 we will obtain a simple representation of S4 since σ is surjective and the invariant subspaces for ρ and ρσ are the same. Thus the simple characters of S3 give a set of simple characters of S4 obtained by applying σ and evaluating the character of S3 . This procedure, which in general works whenever one group is a homomorphic image of another, gives the trivial, sign and 2-dimensional representations of S4 .

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There is an isomorphism between S4 and the group of rotations of R3 which preserve a cube. One sees this from the fact that the group of such rotations permutes the four diagonals of the cube. This action is faithful and since every transposition of diagonals may be realized through some rotation, so can every permutation of the diagonals. Hence the full group of such rotations is isomorphic to S4 . The character of this action of S4 on R3 is the fourth row of the character table. To compute the traces of the matrices which represent the different elements we do not actually have to work out what those matrices are, relying instead on the observation that every rotation of R3 has matrix conjugate to a matrix   cos θ − sin θ 0  sin θ cos θ 0  . 0 0 1

Thus, for example, (12) and (123) must act as rotations through π and 2π 3 , respectively, so act via matrices which are conjugates of √    1  − 2 − 23 0 −1 0 0 √  0 −1 0   3 and − 21 0  2 0 0 1 0 0 1 which have traces −1 and 0. There is an action of S4 on C4 given by the permutation action of S4 on four basis vectors. Since the trace of a permutation matrix equals the number of points fixed by the permutation, this has character χ:

4 2

0 0 1

and we compute

2 1 4 + + 0 + 0 + = 1. 24 4 3 Thus χ = 1 + ψ where ψ is the character of a 3-dimensional representation: hχ, 1i = ψ:

3

1

−1

− 1 0.

Again we have

9 1 1 1 + + + +0 =1 24 4 8 4 so ψ is simple by 3.10, and this is the bottom row of the character table. There are other ways to complete the calculation of the character table. Having computed four of the five rows, the fifth is determined by the facts that it is orthogonal to the other four, and that the sum of the squares of the degrees of the characters equals 24. Equally we could have constructed the bottom row as χ ⊗ ǫ where χ is the other character of degree 3 and ǫ is the sign character. hψ, ψi =

Our next immediate goal is to prove that the character table of a finite group is square, and to deduce the column orthogonality relations. Before doing this we show in the next two results how part of the column orthogonality relations may be derived in a direct way. Consider the regular representation of G on CG, and let χCG denote the character of this representation.

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(3.12) LEMMA. χCG (g) =

n

|G| 0

25

if g = 1 otherwise.

Proof. Each g ∈ G acts by the permutation matrix corresponding to the permutation h 7→ gh. Now χ CG (g) equals the number of 1’s down the diagonal of this matrix, which equals |{h ∈ G gh = h}|. We may deduce an alternative proof of 2.5 (in case k = C), and also a way to do the computation of the final row of the character table once the others have been determined.

(3.13) COROLLARY. Let χ1 , . . . , χr be the simple complex characters of G, with degrees d1 , . . . , dr . Then hχCG , χi i = di , and hence Pr (1) d2i = |G|, and Pi=1 r (2) i=1 di χi (g) = 0 if g 6= 1. Proof. Direct evaluation gives hχCG , χi i =

1 |G|χi (1) = di |G|

and hence χCG = d1 χ1 + · · · + dr χr . Evaluating at 1 gives (1), and at g 6= 1 gives (2). It is an immediate deduction from the fact that the rows of the character table are orthogonal that the number of simple complex characters of a group is at most the number of conjugacy classes of elements in the group. We shall now prove that there is always equality here. For any ring A we denote by Z(A) the center of A. (3.14) LEMMA. (1) For any ring R, Z(Mn (R)) = {λI λ ∈ R} ∼ = R. (2) The number of simple complex characters of G equals dim Z(CG). Proof. (1) Let Eij denote the matrix which is 1 in place i, j and 0 elsewhere. If X = (xij ) is any matrix then Eij X = the matrix with row j of X moved to row i, 0 elsewhere, XEij = the matrix with column i of X moved to column j, 0 elsewhere. If X ∈ Z(Mn (k)) these two are equal, and we deduce that xii = xjj and all other entries in row j and column i are 0. Therefore X = x11 I. (2) In 2.3 we constructed an isomorphism CG ∼ = Mn1 (C) ⊕ · · · ⊕ Mnr (C) where the matrix summands are in bijection with the isomorphism classes of simple modules. On taking centres, each matrix summand contributes 1 to dim Z(CG).

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(3.15) LEMMA. Let x1 , . . . , xt be representatives of the conjugacy classes of elements of G and let R be any ring. For each i let xi ∈ RG denote the sum of the elements in the same conjugacy class as xi . Then Z(RG) is free as an R-module, with basis x1 , . . . , xt . P Proof. We first show that xi ∈ Z(RG). Write xi = y∼xi y, where ∼ denotes conjugacy. Then X X gxi = gy = ( gyg −1 )g = xi g y∼xi

y∼xi

since as y runs through the elements of G conjugate to xi , so does gyg −1, and from this it follows that xi is central. P Next suppose g∈G ag g ∈ Z(RG). We show that if g1 ∼ g2 then ag1 = ag2 . Suppose P P that g2 = hg1 h−1 . The coefficient of g2 in h( g∈G ag g)h−1 is ag1 and in ( g∈G ag g) is ag2 . Since elements of G are independent in RG, these coefficients must be equal. From this we see that every element of Z(RG) can be expressed as an R-linear combination of the xi . Finally we observe that the xi are independent over R, since each is a sum of group elements with support disjoint from the supports of the other xj . (3.16) THEOREM. The number of simple complex characters of G equals the number of conjugacy classes of elements of G. Proof. In 3.14 we showed that the number of simple characters equals the dimension of the centre, and 3.15 we showed that this is equal to the number of conjugacy classes. We conclude that the character table of a finite group is always square. From this we get orthogonality relations between the columns of the character table. (3.17) COROLLARY (Column orthogonality relations). Let X be the character table of G as a matrix, and let  |C (x )|  0 ··· 0 G

 C= 

1

0 .. .

|CG (x2 )|

  ..  . . · · · |CG (xr )| ..

0

where x1 , . . . , xr are representatives of the conjugacy classes of elements of G. Then X T X = C, where the bar denotes complex conjugation. Proof. The orthogonality relations between the rows may be stated XC −1 X T = I. Since all these matrices are square, they are invertible, and in fact (XC −1 )−1 = X T = −1 CX . Therefore X T X = C.

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A less intimidating way to state the column orthogonality relations is r X i=1

χi (g)χi (h) =



|CG (g)| if g ∼ h, 0 if g 6∼ h.

The special case of this in which g = 1 has already been seen in Corollary 3.13. We conclude this section on the properties of characters with the result which states that the degrees of the simple complex characters of a finite group G divide |G|. This is, of course, a big restriction on the possible degrees which may occur, and it is proved by means of an algebraic excursion which otherwise will not be used in the immediate development of the theory. This means that it is quite a good idea to skip the proof on first reading. We need to use the notion of integrality. Suppose that S is a commutative ring with 1 and R is a subring of S with the same 1. An element s ∈ S is said to be integral over R if f (s) = 0 for some monic polynomial f ∈ R[X]. Here, a monic polynomial is one in which the coefficient of the highest power of X is 1. We say that the ring S is integral over R if every element of S is integral over R. An element of C integral over Z is called an algebraic integer. We summarize the properties of integers which we will need.

(1)

(2) (3) (4)

(3.18) THEOREM. Let R be a subring of a commutative ring S. The following are equivalent for an element s ∈ S: (a) s is integral over R, (b) R[s] is contained in some finitely generated R-submodule M of S such that sM ⊆ M. The elements of S integral over R form a subring of S. {x ∈ Q x is integral over Z} = Z. Let g be any element of a finite group G and χ any character of G. Then χ(g) is an algebraic integer. In the statement of this result, R[s] denotes the subring of S generated by R and s. Proof. (1) (a) ⇒ (b). Suppose s is an element integral over R, satisfying the equation sn + an−1 sn−1 + · · · + a1 s + a0 = 0

where ai ∈ R. Then R[s] is generated as an R-module by 1, s, s2 , . . . , sn−1 . This is because the R-span of these elements is also closed under multiplication by s, using the fact that s · sn−1 = −an−1 sn−1 − · · · − a1 s − a0 . Thus we may take M = R[s]. (b) ⇒ (a). Suppose R[s] ⊆ M = Rx1 + · · · Rxn with sM ⊆ M . Thus for each i we Pn have sxi = j=1 λij xj for certain λij ∈ R. Consider the n × n matrix A = sI − (λij ) with entries in S, where I is the identity matrix. We have Ax = 0 where x is the vector

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(x1 , . . . , xn )T , and so adj(A)Ax = 0 where adj(A) is the adjugate matrix of A satisfying adj(A)A = det(A) · I. Hence det(A) · xi = 0 for all i. Since 1 ∈ R ⊆ M is a linear combination of the xi we have det(A) = 0 and so s is a root of the monic polynomial det(X · I − (λij )). (2) We show that if a, b ∈ S are integral over R then a + b and ab are also integral over R. These lie in R[a, b], and we show that this is finitely generated as an R-module. We see from the proof of part (1) that each of R[a] and R[b] is finitely generated as an R-module. If R[a] is generated by x1 , . . . , xm and R[b] is generated by y1 , . . . , yn , then R[a, b] is evidently generated as an R-module by all the products xi yj . Now R[a, b] also satisfies the remaining condition of part (b) of (1), and we deduce that a + b and ab are integral over R. (3) Suppose that ab is integral over Z, where a, b are coprime integers. Then  a n b

+ cn−1

 a n−1 b

+ · · · + c1

a + c0 = 0 b

for certain integers ci , and so an + cn−1 an−1 b + · · · + c1 abn−1 + c0 bn = 0. Since b divides all terms in this equation except perhaps the term an , b must also be a factor of an . Since a and b are coprime, this is only possible if b = ±1, and we deduce that a ∈ Z. b (4) χ(g) is the sum of the eigenvalues of g in its action on the representation which affords χ. Since g n = 1 for some n these eigenvalues are all roots of X n − 1 and so are integers. (3.19) PROPOSITION. The centre Z(ZG) is integral over Z. Hence if x1 , . . . , xr are representatives of the conjugacy classes of G, xi ∈ ZG is the sum of the elements conjugate Pr to xi , and λ1 , . . . , λr ∈ C are algebraic integers then i=1 λi xi is integral over Z. Proof. In the statement of this result we are identifying Z with the scalar multiples of the identity Z · 1 ⊆ Z(ZG). We also regard ZG as a subset of CG when we form the Pr linear combination i=1 λi xi . It is the case that every commutative subring of ZG is integral over Z, using condition 1(b) of 3.18, since such a subring is in particular a subgroup of the finitely-generated free abelian group ZG, and hence is finitely generated as a Z-module. We have seen in 3.15 that the elements x1 , . . . , xr lie in Z(ZG), so they are integral Pr over Z, and by part (2) of 3.18 the linear combination i=1 λi xi is integral also. (We note that the xi are in fact a finite set of generators for Z(ZG) as an abelian group, but we did not need to know this for the proof.)

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Let ρ1 , . . . , ρr be the simple representations of G over C with degrees d1 , . . . , dr and characters χ1 , . . . , χr . Then each ρi : G → Mdi (C) extends by linearity to a C-algebra homomorphism r M ρi : CG = Mdi (C) → Mdi (C) i=1

projecting onto the ith matrix summand. The fact that the group homomorphism ρi extends to an algebra homomorphism in this way comes formally from the construction of the group algebra. The fact that this algebra homomorphism is projection onto the ith summand arises from the way we decomposed CG as a sum of matrix algebras, in which each matrix summand acts on the corresponding simple module as matrices on the space of column vectors. (3.20) PROPOSITION. If x ∈ Z(CG) then ρi (x) = λI for some λ ∈ C. In fact ρi (x) = Writing x =

P

g∈G ag g

1 · trace of ρi (x) · I. di

we have ρi (x) =

1 X ag χi (g) · I. di g∈G

Proof. Since x is central the matrix ρi (x) commutes with the matrices ρi (g) for all g ∈ G. Therefore by Schur’s lemma, since ρi is a simple complex representation, ρi (x) = λI, some scalar multiple of the identity matrix. Evidently λ = d1i tr(λI). Substituting into this expression we obtain X 1 1 X tr(ρi ( ag g)) = ag tr(ρi (g)) di di g∈G

g∈G

1 X ag χi (g). = di g∈G

(3.21) THEOREM. The degrees di of the simple complex representations of G all divide |G|. P Proof. Let x = g∈G χi (g −1 )g. This element is central in CG since the coefficients of group elements are constant on conjugacy classes. By 3.20 1 X ρi (x) = χi (g −1 )χi (g) · I di g∈G

=

|G| ·I di

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the second equality arising from the fact that hχi , χi i = 1. Now x is integral over Z · 1 so ρi (x) is integral over ρi (Z · 1) = Z · I. Thus |G| is integral over Z, hence |G| ∈ Z so di di di |G|.

At this point we may very quickly deduce a formula for the central primitive idempotent elements in CG. We use some of the same ideas that arose in proving Theorem 3.21 and for that reason we present this further application here. However, the formula has no immediate use in this text and can be omitted without loss of understanding at this point. We start with some generalities about central primitive idempotent elements. An element e of an algebra A is said to be idempotent if e2 = e, and we say it is a central idempotent element if it lies in the centre Z(A). Two idempotent elements e and f are orthogonal if ef = f e = 0. An idempotent element e is called primitive if whenever e = e1 + e2 where e1 and e2 are orthogonal idempotent elements then either e1 = 0 or e2 = 0. We say that e is a primitive central idempotent element if it is primitive as an idempotent element in Z(A), that is, e is central and has no proper decomposition as a sum of orthogonal central idempotent elements. Some properties of idempotent elements relating to these definitions are described in the exercises to Section 2. Given a set of rings with identity A1 , . . . , Ar we may form their direct sum A = A1 ⊕ · · · ⊕ Ar which acquires the structure of a ring with componentwise addition and multiplication. In this situation each ring Ai may be identified as the subset of A consisting of elements which are zero except in component i, but this subset is not a subring of A because it does not contain the identity element of A. It is, however, a 2-sided ideal. Equally, in any decomposition of a ring A as a direct sum of 2-sided ideals, these ideals have the structure of rings with identity. (3.22) PROPOSITION. Let A be a ring with identity. Decompositions A = A1 ⊕ · · · ⊕ Ar as direct sums of 2-sided ideals Ai biject with expressions 1 = e1 + · · · + er as a sum of orthogonal central idempotent elements, where ei is the identity element of Ai and Ai = Aei . The Ai are indecomposable as rings if and only if the ei are primitive central idempotent elements. If every Ai is indecomposable as a ring then the Ai , and also the primitive central idempotents ei , are uniquely determined as subsets of A, and every central idempotent can be written as a sum of certain of the ei . Proof. Given any ring decomposition A = A1 ⊕ · · ·⊕ Ar we may write 1 = e1 + · · ·+ er where ei ∈ Ai and now it is clear that the ei are orthogonal central idempotent elements. Conversely, given an expression 1 = e1 + · · · + er where the ei are orthogonal central idempotent elements we have A = Ae1 ⊕ · · · ⊕ Aer as rings.

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To say that the ring Ai is indecomposable means that it cannot be expressed as a direct sum of rings, except in the trivial way, and evidently this happens precisely if the corresponding idempotent element cannot be decomposed as a sum of orthogonal central idempotent elements. What is perhaps surprising is that there is at most one decomposition of A as a sum of indecomposable rings. Suppose we have two such decompositions, and that the corresponding primitive central idempotent elements are labelled ei and fj , so that 1 = e1 + · · · + er = f1 + · · · + fs . We have ei = ei · 1 =

s X

ei fj ,

j=1

and so ei = ei fj for some unique j and ei fk = 0 if k 6= j, by primitivity of ei . Also fj = 1 · fj =

r X

ek fj

k=1

so that ek fj 6= 0 for some unique k. Since ei fj 6= 0 we have k = i and ei fj = fj . Thus ei = fj . We proceed by induction on r, starting at r = 1. If r > 1 we now work with the P P ring A · k6=i ek = A · k6=j fk in which the identity is expressible as sums of primitive P P central idempotent elements k6=i ek = k6=j fk . The first of these expressions has r − 1 terms, so by induction the ek ’s are the same as the fk ’s after some permutation. If e is any central idempotent and the ei are primitive then eei is either ei or 0 since e = eei + e(1 − ei ) is a sum of orthogonal central idempotents. Thus e=e

r X

ei =

k=1

r X

eei

k=1

is a sum of certain of the ei . In view of the last result when we speak of the primitive central idempotent elements of an algebra we are referring to the set which determines the unique decomposition of the algebra as a direct sum of indecomposable rings. (3.23) THEOREM. Let χ1 , . . . , χr be the simple complex characters of G with degrees d1 , . . . , dr . The primitive central idempotent elements in CG are the elements di X χi (g −1 )g |G| g∈G

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where 1 ≤ i ≤ r, the corresponding indecomposable ring summand of CG having a simple representation which affords the character χi . Proof. Using the notation of 3.20 we have that the representation ρi which affords χi yields an algebra map ρi : CG → Mdi (C) which is projection onto the ith matrix summand in a decomposition of CG as a sum of matrix rings. For any field k the matrix ring Mn (k) is indecomposable, since we have seen in 3.14 that Z(Mn (k)) ∼ = k and the only non-zero idempotent element in a field is 1. Thus the decomposition of CG as a direct sum of matrix rings is the unique decomposition of CG as a sum of indecomposable ring summands. The corresponding primitive central idempotent elements are the identity matrices in the various summands, and so they are the elements ei ∈ CG such that ρi (ei ) = I and ρj (ei ) = 0 if i 6= j. From the formula of 3.20 and the orthogonality relations we have ρj (

di X di X χi (g −1 )g) = χi (g −1 )χj (g) · I |G| |G|dj g∈G

g∈G

di hχi , χj i · I dj di = δi,j · I dj =

= δi,j · I, so that the elements specified in the statement of the theorem do indeed project correctly onto the identity matrices, and are therefore the primitive central idempotent elements. While the identity matrix is a primitive central idempotent element in the matrix ring Mn (k), where k is a field, it is never a primitive idempotent element if n > 1 since it is the sum of the orthogonal (non-central) primitive idempotent elements I = E1,1 + · · · + En,n . Furthermore, removing the hypothesis of centrality we can no longer say that decompositions of the identity as a sum of primitive idempotent elements are unique; indeed, any conjugate expression by an invertible matrix will also be a sum of orthogonal primitive idempotent elements. Applying these comments to a matrix summand of CG, the primitive idempotent decompositions of 1 will never be unique if we have a non-abelian matrix summand — which, of course, happens precisely when G is non-abelian. It is unfortunately the case that in terms of the group elements there is in general no known formula for primitive idempotent elements of CG lying in a non-abelian matrix summand. We conclude this section with Burnside’s remarkable ‘pa q b theorem’, which establishes a group-theoretic result using the ideas of representation theory we have so far developed, together with some admirable ingenuity. In the course of the proof we again make use of the idea of integrality, but this time we also require Galois theory at one point. This is needed to show that if ζ is a field element which is expressible as a sum of roots of unity,

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then every algebraic conjugate of ζ is again expressible as a sum of roots of unity. We present Burnside’s theorem here because of its importance as a theorem in its own right, not because anything later depends on it. In view of this the proof (which is fairly long) can be omitted without subsequent loss of understanding. Recall that a group G is soluble if it has a composition series in which all of the composition factors are cyclic. Thus a group is not soluble precisely if it has a non-abelian composition factor. (3.24) THEOREM (Burnside’s pa q b theorem). Let G be a group of order pa q b where p and q are primes. Then G is soluble. Proof. We suppose the result is false, and consider a group G of minimal order subject to being not soluble and of order pa q b . Step 1. The group G is simple, not abelian and not of prime-power order; for if it were abelian or of prime-power order it would be soluble, and if G had a normal subgroup N then one of N and G/N would be a smaller group of order pα q β which was not soluble. Step 2. We show that G contains an element g whose conjugacy class has size q d for some d > 0. Let P be a Sylow p-subgroup, 1 6= g ∈ Z(P ). Then CG (g) ⊇ P so |G : CG (g)| = q d for some d > 0, and this is the number of conjugates of g. Step 3. We show that there is a simple non-identity character χ of G such that q6 χ(1) and χ(g) 6= 0. To prove this, suppose to the contrary that whenever χ 6= 1 and q 6 χ(1) then χ(g) = 0. Let R denote the ring of algebraic integers in C. Consider the orthogonality relation between the column of 1 (consisting of character degrees) and the column of g: X 1+ χ(1)χ(g) = 0. χ6=1

Then q divides every term apart from 1 in the sum on the left, and so 1 ∈ qR. Thus −1 q −1 ∈ R. But q −1 ∈ Q and so q ∈ Z by 3.18, a contradiction. We now fix a non-identity character χ for which q6 χ(1) and χ(g) 6= 0. Step 4. Recall that the number of conjugates of g is q d . We show that q d χ(g) χ(1) is an algebraic integer. To do this we use results 3.15 and 3.20. These imply that if P g = h∼g h ∈ CG is the sum of the elements conjugate to g and ρ is a representation affording the character χ then g ∈ Z(CG) and ρ(g) =

1 X χ(h) · I χ(1) h∼g

d

=

q χ(g) · I, χ(1)

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where I is the identity matrix. Now by 3.19 this is integral over ρ(Z) = Z · I, which proves what we want. Step 5. We deduce that χ(g) χ(1) is an algebraic integer. This arises from the fact that q6 χ(1). We can find λ, µ ∈ Z so that λq d + µχ(1) = 1. Now χ(g) q d χ(g) =λ + µχ(g) χ(1) χ(1)

is a sum of algebraic integers. Step 6. We show that |χ(g)| = χ(1) and put ζ = χ(g)/χ(1). We consider the algebraic conjugates of ζ, which are the roots of the minimal polynomial of ζ over Q. They are all algebraic integers, since ζ and its algebraic conjugates are all roots of the same polynomials over Q. Thus the product N (ζ) of the algebraic conjugates is an algebraic integer. Since it is also ± the constant term of the minimal polynomial of ζ, it is rational and non-zero. Therefore 0 6= N (ζ) ∈ Z by 3.18. Now χ(g) is the sum of the eigenvalues of χ(g), of which there are χ(1), each of which is a root of unity. Hence by the triangle inequality, |χ(g)| ≤ χ(1). By Galois theory the same is true and a similar inequality holds for each algebraic conjugate of χ(g). We conclude that all algebraic conjugates of ζ have absolute value at most 1. Therefore |N (ζ)| ≤ 1. The only possibility is |N (ζ)| = 1 and |ζ| = 1. Step 7. We deduce that G has a proper normal subgroup by considering H = {h ∈ G |χ(h)| = χ(1)}

where χ is the simple non-identity character introduced in Step 3. We argue first that H is a normal subgroup. If the eigenvalues of ρ(h) are λ1 , . . . , λn then, since these are roots of unity, |λ1 + · · · + λn | = n if and only if λ1 = · · · = λn . Thus |χ(h)| = χ(1) if and only if ρ(h) is multiplication by some scalar, and from this we see immediately that H is a normal subgroup. It also implies that H/ Ker ρ is abelian. From Step 6 we see that |H/ Ker ρ| > 1, but this forces a contradiction since simplicity of G implies that H = G and Ker ρ = 1, from which we deduce that G is abelian. This contradiction terminates the proof. Exercises for Section 3. 1. Suppose that V is a representation of G over C for which χV (g) = 0 if g 6= 1. Show that dim V is a multiple of |G|. Deduce that V ∼ = CGn for some n. Show that if W is any representation of G over C then CG ⊗C W ∼ = CGdim W as CG-modules. 2. (a) By using characters show that if V and W are any CG-modules then (V ⊗C ∗ ∼ W ) = V ∗ ⊗C W ∗ , and (CG CG)∗ ∼ = CG CG. (b) If k is any field and V , W are kG-modules, show that (V ⊗k W )∗ ∼ = V ∗ ⊗k W ∗ , and (kG kG)∗ ∼ = kG kG.

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3. Consider a ring with identity which is the direct sum (as a ring) of subrings A = A1 ⊕ · · · ⊕ Ar . Suppose that A has exactly n isomorphism types of simple modules. Show that r ≤ n. 4. Let g be any non-identity element of a group G. Show that G has a simple complex character χ for which χ(g) has negative real part. 5. Show that if every element of a finite group G is conjugate to its inverse, then every character on G is real-valued. Conversely, show that if every character on G is real-valued, then every element of G is conjugate to its inverse. [Note here that the quaternion group of order 8 in its action on the algebra of quaternions provides an example of a complex representation which is not equivalent to a real representation, but whose character is real-valued. In this example, the representation has complex dimension 2, but there is no basis over C for the representation space such that the group acts by matrices with real entries.] 6. (Jozsef Pelikan) While walking down the street you find a scrap of paper with the following character table on it: 1 1 1 −1 · · · 2 · · · −1 · · · 3 1 3 −1 All except two of the columns are obscured, and while it is clear that there are five rows you cannot read anything of the other columns, including their position. Prove that there is an error in the table. Given that there is exactly one error, determine where it is, and what the correct entry should be. 7. A finite group has seven conjugacy classes with representatives c1 , . . . , c7 (where c1 = 1), and the values of five of its irreducible characters are given by the following table: c1 c2 c3 c4 c5 c6 c7 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 4 1 −1 0 2 −1 0 4 1 −1 0 −2 1 0 5 −1 0 1 1 1 −1 Calculate the numbers of elements in the various conjugacy classes and the remaining simple characters. 8. Let g ∈ G. Prove that g lies in the centre of G if and only if |χ(g)| = |χ(1)| for every simple complex character χ of G.

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9. Here is a column of a character table: g 1 −1 0 −1 −1√

−1+i 11 2√ −1−i 11 2

0 1 0

(a) Find the order of g. (b) Prove that g 6∈ Z(G). (c) Show that there exists an element h ∈ G with the same order as g but not conjugate to g. (d) Show that there exist two distinct simple characters of G of the same degree.

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4. The Construction of Modules and Characters We start with the particular case of cyclic groups over C. (4.1) PROPOSITION. Let G = hx xn = 1i be a cyclic group of order n, and let ζ ∈ C be a primitive nth root of unity. Then the simple complex characters of G are the n functions χr (xs ) = ζ rs where 0 ≤ r ≤ n − 1. Proof. We merely observe that the mapping xs 7→ ζ rs is a group homomorphism G → GL(1, C) = C∗ giving a 1-dimensional representation with character χr , which must necessarily be simple. These characters are all distinct, and since the number of them equals the group order we have them all. We next show how to obtain the simple characters of a product of groups in terms of the characters of the groups in the product. Combining this with the last result we obtain the character table of any finite abelian group. We describe a construction which works over any ring R. Suppose that ρ1 : G1 → GL(V1 ) and ρ2 : G2 → GL(V2 ) are representations of groups G1 and G2 . We may define an action of G1 × G2 on V1 ⊗R V2 by the formula (g1 , g2 )(v1 ⊗ v2 ) = g1 v1 ⊗ g2 v2 where gi ∈ Gi and vi ∈ Vi . When R is a field we may choose bases for V1 and V2 , and now (g1 , g2 ) acts via the tensor product of the matrices by which g1 and g2 act. It follows when R = C that χV1 ⊗V2 (g1 , g2 ) = χV1 (g1 )χV1 (g1 ). (4.2) THEOREM. Let V1 , . . . , Vm and W1 , . . . , Wn be complete lists of the simple complex representations of groups G1 and G2 . Then the representations Vi ⊗ Wj with 1 ≤ i ≤ n and 1 ≤ j ≤ m form a complete list of the simple complex G1 × G2 representations. Remark. The statement of this theorem actually works over an arbitrary field, which need not be C. There is a still more general statement to do with representations of finitedimensional algebras A and B over a field k, which is that the simple representations of A⊗k B are precisely the S ⊗k T , where S is a simple A-module and T is a simple B-module. The connection with groups is that the group algebra R[G1 × G2 ] over any commutative

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ring R is isomorphic to RG1 ⊗R RG2 , which we may see by observing that the basis of RG1 ⊗R RG2 consisting of elements g1 ⊗ g2 with gi ∈ Gi multiplies the same way as the group G1 × G2 . Proof. We first verify that the representations Vi ⊗ Wj are simple using the criterion of 3.10: X 1 hχVi ⊗Wj , χVi ⊗Wj i = χVi ⊗Wj (g1 , g2 )χVi ⊗Wj (g1 , g2 ) |G1 × G2 | = =

1 |G1 ||G2 | 1 |G1 |

(g1 ,g2 )∈G1 ×G2

X

χV1 (g1 )χV1 (g1 )χV2 (g2 )χV2 (g2 )

(g1 ,g2 )∈G1 ×G2

X

χV1 (g1 )χV1 (g1 ) ·

g1 ∈G1

1 X χV2 (g2 )χV2 (g2 ) |G2 | g2 ∈G2

= 1. The characters of these representations are distinct, since by a similar calculation if (i, j) 6= (r, s) then hχVi ⊗Wj , χVr ⊗Ws i = 0. To show that we have the complete list, we observe that if dim Vi = di and dim Wj = ej then Vi ⊗ Wj is a representation of degree di ej and n m m X n X X X 2 2 e2j = 1. di · (di ej ) = i=1

i=1 j=1

j=1

This establishes what we need, using 2.5 or 3.13. Putting the last two results together enables us to compute the character table of any finite abelian group. To give a very small example, let G = hx, y x2 = y 2 = [x, y] = 1i ∼ = C2 × C2 . The character tables of hxi and hyi are

χ1 χ2

2 2 1 x 1 1 1 −1

ψ1 ψ2

2 2 1 y 1 1 1 −1

TABLE: Two copies of the character table of C2 . and the character table of C2 × C2 is χ1 ψ1 χ2 ψ1 χ1 ψ2 χ2 ψ2

4 4 4 4 1 x y xy 1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1

TABLE: The character table of C2 × C2 . We immediately notice that this construction gives the character table of C2 × C2 as the tensor product of the character tables of C2 and C2 , and without further argument we see that this is true in general.

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(4.3) COROLLARY. The character table of a direct product G1 × G2 is the tensor product of the character tables of G1 and G2 . We see from this that all simple complex characters of an abelian group have degree 1, and in fact this property characterizes abelian groups. (4.4) THEOREM. The following are equivalent for a finite group G: (1) G is abelian, (2) all simple complex representations of G have degree 1. Proof. Since the simple representations of every finite cyclic group all have degree 1, and since every finite abelian group is a direct product of cyclic groups, the last result shows that all simple representations of a finite abelian group have degree 1. Pr Conversely, we may use the fact that |G| = i=1 d2i where d1 , . . . , dr are the degrees of the simple representations. We deduce that di = 1 for all i ⇔ r = |G| ⇔ every conjugacy class has size 1 ⇔ every element is central ⇔ G is abelian. Another proof of this result may be obtained from the fact that CG is a direct sum of matrix algebras over C, a summand Mn (C) appearing precisely if there is a simple module of dimension n. The group and hence the group ring are abelian if and only if n is always 1. We may construct the part of the character table of any finite group which consists of characters of degree 1 by combining the previus results with the next one. (4.5) PROPOSITION. The degree 1 representations of any finite group G over any field are precisely the degree 1 representations of G/G′ , lifted to G via the homomorphism G → G/G′ . Proof. We only have to observe that a degree 1 representation of G over a field k is a homomorphism G → GL(1, k) = k × which takes values in an abelian group, and so has kernel containing G′ . Thus such a homomorphism is always a composite G → G/G′ → GL(1, k) obtained from a degree 1 representation of G/G′ . (4.6) Example. Neither implication of 4.4 holds if we do not assume that our representations are defined over C (or more generally, 3an algebraically closed field in characteristic prime to |G|). Over R the cyclic group hx x = 1i of order 3 has a 2-dimensional repre. This representation is simple since there sentation in which x acts as rotation through 2π 3 is no 1-dimensional subspace stable under the group action. We need to pass to C to split it as a sum of two representations of degree 1. This is an example of an abelian group not all of whose simple representations have degree 1, and equally one may find a non-abelian group all of whose simple representations do have degree 1. As we shall see later, this happens whenever G is a p-group and we consider representations in characteristic p.

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Induction and Restriction We now consider an extremely important way of constructing representations of a group from representations of its subgroups. Let H be a subgroup of G and V an RHmodule where R is a commutative ring with 1. We define an RG-module V ↑G H = RG ⊗RH V with the action of G coming from the left module action on RG: X X x·( ag g ⊗ v) = (x ag g) ⊗ v g∈G

g∈G

where x, g ∈ G, ag ∈ R and v ∈ V . We refer to this module as V induced from H to G, G and say that V ↑G is used for this H is an induced module. In many books the notation V induced module, but for us this conflicts with the notation for fixed points. We denote the set of left cosets {gH g ∈ G} by G/H.

(4.7) PROPOSITION. Let V be an RH-module and let g1 H, . . . , g|G:H| H be a list of the left cosets G/H. Then |G:H| M G gi ⊗ V V ↑H = i=1

as R-modules, where gi ⊗ V = {gi ⊗ v v ∈ V } ⊆ RG ⊗RH V . Each gi ⊗ V is isomorphic to V as an R-module, and in case V is free as an R-module we have rankR V ↑G H = |G : H| rankR V. If x ∈ G then x(gi ⊗V ) = gj ⊗V where xgi = gj h for some h ∈ H. Thus the R-submodules gi ⊗V of V ↑G H are permuted under the action of G. This action is transitive, and if g1 ∈ H then StabG (g1 ⊗ V ) = H. L|G:H| |G:H| ∼ Proof. We have RGRH = as right RH-modules, since H i=1 gi RH = RH has a permutation action on the basis of RG with |G : H| orbits g1 H, . . . , g|G:H| H, and each orbit spans a right RH-submodule R[gi H] of RG, which is isomorphic to RHRH as right RH-modules via the isomorphism specified by gi h 7→ h, where h ∈ H. Now RG ⊗RH V = (

|G:H|

M

gi RH) ⊗RH V

i=1

=

|G:H|

M

(gi RH ⊗RH V )

i=1

=

|G:H|

M i=1

gi ⊗RH V

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and as R-modules gi RH ⊗RH V ∼ = RH ⊗RH V ∼ =V. We next show that with its left action on RG ⊗RH V coming from the left action on RG, G permutes these R-submodules. If x ∈ G and xgi = gj h with h ∈ H then x(gi ⊗ v) = xgi ⊗ v = gj h ⊗ v = gj ⊗ hv, so that x(gi ⊗ v) ⊆ gj ⊗ V . We argue that we have equality using the invertibility of x. For, by a similar argument to the one above, we have x−1 gj ⊗ V ⊆ gi ⊗ V , and so gj ⊗ V = xx−1 (gj ⊗ V ) ⊆ x(gi ⊗ V ). This action of G on the subspaces is transitive since given two subspaces gi ⊗ V and gj ⊗ V we have (gj gi−1 )gi ⊗ V = gj ⊗ V . Now to compute the stabilizer of g1 ⊗ V where g1 ∈ H, if x ∈ H then x(g1 ⊗ V ) = g1 (g1−1 xg1 ) ⊗ V = g1 ⊗ V , and if x 6∈ H then x ∈ gi H for some i 6= 1 and so x(g1 ⊗ V ) = gi ⊗ V . Thus StabG (g1 ⊗ V ) = H. The structure of induced modules described in the last result in fact characterizes these modules, giving an extremely useful criterion for a module to be of this form which we will use several times later on. (4.8) PROPOSITION. Let M be an RG-module which has an R-submodule V with the property that M is the direct sum of the R-submodules {gV g ∈ G}. Let H = {g ∈ G gV = V }. Then M ∼ = V ↑G . H

Proof. We define a map of R-modules RG ⊗RH V → M g ⊗ v 7→ gv extending this specification from the generators to the whole of RG ⊗RH V by R-linearity. This is in fact a map of RG-modules. The R-submodules gV of M are in bijection with the cosets G/H, since G permutes them transitively, and the stabilizer of one of them is H. Thus each of RG ⊗RH V and M is the direct sum of |G : H| R-submodules g ⊗ V and gV respectively, each isomorphic to V via isomorphisms g ⊗ v ↔ v and gv ↔ v. Thus on each summand the map g ⊗ v 7→ gv is an isomorphism, and so RG ⊗RH V → M is itself an isomorphism.

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(4.9) Examples. 1. Immediately from the definitions we have RG = R ↑G 1. 2. More generally, suppose that Ω is a G-set, that is a set with an action of G. We may form RΩ, the free R-module with the elements of Ω as a basis, and it acquires the structure of an RG-module via the permutation action of G on this basis. This is the L permutation module determined by Ω. Now RΩ = ω∈Ω Rω is the direct sum of rank 1 R-submodules, each generated by a basis vector. In case G acts transitively on Ω these are permuted transitively by G. If we pick any ω ∈ Ω and let H = Stab ω then H is also the stabilizer of the space Rω and RΩ ∼ = R ↑G H . This shows that permutation modules on transitive G-sets are induced modules. The general function of induced representations is that they are a mechanism which relates the representations of a group to those of a subgroup. When working over C they provide a way of constructing new characters, and with this in mind we give the formula for the character of an induced representation. If χ is the character of a representation V G of a subgroup H, let us simply write χ ↑G H for the character of V ↑H . We will also write [G/H] to denote a set of representatives of the left cosets of H in G. (4.10) PROPOSITION. Let H be a subgroup of G and let V be a CH-module with character χ. Then the character of V ↑G H is χ ↑G H (g) = =

1 |H|

X

χ(t−1 gt)

t∈G t−1 gt∈H

X

χ(t−1 gt).

t∈[G/H] t−1 gt∈H

Proof. The two formulas on the right are in fact the same, since if t−1 gt ∈ H and h ∈ H then (th)−1 gth ∈ H also, and so {t ∈ G t−1 gt ∈ H} is a union of left cosets of H. Since χ(t−1 gt) = χ((th)−1 gth) the terms in the first sum are constant on the cosets of H, and we obtain the second sum by choosing one representative from each coset and multiplying by |H|. Using the vector space decomposition of 4.7 we obtain that the trace of g on V ↑G H is the sum of the traces of g on the spaces t⊗V which are invariant under g, where t ∈ [G/H]. This is because if g does not leave t ⊗ V invariant, we get a matrix of zeros on the diagonal at that point in the block matrix decomposition for the matrix of g. Thus we only get a non-zero contribution from subspaces t ⊗ V with gt ⊗ V = t ⊗ V . This happens if and only if t−1 gt ⊗ V = 1 ⊗ V , that is t−1 gt ∈ H. We have χ ↑G H (g) =

X

t∈[G/H] t−1 gt∈H

trace of g on t ⊗ V.

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Now g acts on t ⊗ V as g(t ⊗ v) = t(t−1 gt) ⊗ v = t ⊗ (t−1 gt)v and so the trace of g on this space is χ(t−1 gt). Combining this with the last expression gives the result. We see in the above proof that g leaves invariant t ⊗ V if and only if t−1 gt ∈ H, or in other words g ∈ tHt−1 . Thus StabG (t ⊗ V ) = tHt−1 . Furthermore, if we identify t ⊗ V with V by means of the bijection t ⊗ v ↔ v, then g acts on t ⊗ V via the composite homomorphism c

−1

ρ

t hgi−→H −→GL(V )

where ρ is the homomorphism associated to V and ca (x) = axa−1 is the automorphism of G which is conjugation by a ∈ G. (4.11) Example. To make clearer what the terms in the expression for the induced character are, consider G = S3 and H = h(123)i the normal subgroup of order 3. To avoid expressions such as (()) we will write the identity element of S3 as e. We may take the coset representatives [G/H] to be {e, (12)}. If χ is the trivial character of H then e (12) χ ↑G )=2 H (e) = χ(e ) + χ(e

χ ↑G H ((12)) = the empty sum = 0 e (12) χ ↑G )=2 H ((123)) = χ((123) ) + χ((123)

Recalling the character table of S3 we find that χ ↑G H is the sum of the trivial character and the sign character of S3 . Before giving examples of how induced characters may be used in the construction of character tables, we describe some formalism of the relationship between a group and its subgroups which will allow us to compute more easily with induced representations. The companion notion to induction is that of restriction of representations. If H is a subgroup of G and W is a representation of G we denote by W ↓G H the representation of H obtained by letting the elements of H the way they do when regarded as elements of G. Restriction and induction are a particular case of the following more general situation. Whenever we have a (unital) homomorphism of rings A → B, an A-module V and an B-module W we may form the B-module B ⊗A V and the A-module W ↓B A . On taking A = RH and B = RG we obtain the induction and restriction we have been studying. (4.12) LEMMA. Let A → B be a homomorphism of rings, V an A-module and W a B-module. (1) (Adjointness of ⊗ and Hom) HomB (B ⊗A V, W ) ∼ = HomA (V, W ↓A ).

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(2) If φ : B → C is another ring homomorphism then C ⊗B (B ⊗A V ) ∼ = C ⊗A V . Proof. In the case of (1) the mutually inverse isomorphisms are f 7→ (v 7→ f (1 ⊗ v)) and (b ⊗ v 7→ bg(v)) ← g. In the case of (2) the mutually inverse isomorphisms are c ⊗ b ⊗ v 7→ cφ(b) ⊗ v and c ⊗ 1 ⊗ v ← c ⊗ v.

(4.13) COROLLARY. Let H ≤ K ≤ G be subgroups of G, let V be an RH-module and W an RG-module. G ∼ (1) (Frobenius reciprocity) HomRG (V ↑G H , W ) = HomRH (V, W ↓H ). (2) (Transitivity of induction) (V ↑K ) ↑G ∼ = V ↑G . (3) (4)

H K G (Transitivity of restriction) (W ↓K ) ↓K H= G G G V ↑H ⊗R W ∼ = (V ⊗R W ↓H ) ↑H .

H

W ↓G H.

Proof. The first two are the translation of 4.12 into the language of group representations and the third statement is clear. Part (4) is the statement (RG ⊗RH V ) ⊗R W ∼ = RG ⊗RH (V ⊗R W ) and is not a corollary of 4.12. Here the mutually inverse isomorphisms are (g ⊗ v) ⊗ w 7→ g ⊗ (v ⊗ g −1 w) and (g ⊗ v) ⊗ gw ← g ⊗ (v ⊗ w).

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In the case of representations in characteristic zero all of these results may be translated into the language of characters. By analogy with the notation χ ↑G K for the character G K of an induced representation V ↑K , let us write χ ↓H for the character of V ↓K H. (4.14) COROLLARY. Let H ≤ K ≤ G be subgroups of G, let χ be a complex character of H and ψ a character of G. (1) (Frobenius reciprocity) G hχ ↑G H , ψiG = hχ, ψ ↓H iH and G hψ, χ ↑G H iG = hψ ↓H , χiH .

In fact all four numbers are equal. G G (2) (Transitivity of induction) (χ ↑K H ) ↑K = χ ↑H . K G (3) (Transitivity of restriction) (ψ ↓G K ) ↓H = ψ ↓H . G G (4) χ ↑G H ·ψ = (χ · ψ ↓H ) ↑H . Proof. In (1) we write h , iG and h , iH to denote the inner product of characters of G and H, respectively. The four parts are translations of the four parts of 4.13 into the language of characters. In part (1) we use the fact that the inner products are the dimensions of the Hom groups in 4.13(1) and that the inner product is symmetric. The statement of Frobenius reciprocity for complex characters is equivalent to the statement that if ψ and χ are simple characters of G and H respectively then the multiG plicity of ψ as a summand of χ ↑G H equals the multiplicity of χ as a summand of ψ ↓H . At a slightly more sophisticated level we may interpret induction, restriction and Frobenius reciprocity in terms of the space Ccc(G) of class functions introduced in Section 3, that is, the vector space of functions cc(G) → C where cc(G) is the set of conjugacy classes of G. Since each conjugacy class of H is contained in a unique conjugacy class of G we have a mapping cc(H) → cc(G) and this gives rise by composition to a linear cc(G) map ↓G → Ccc(H) which on characters is the restriction operation we have already H: C cc(H) defined. We may also define a linear map ↑G → Ccc(G) which on characters sends H: C a character χ of H to the character χ ↑G H . It would be possible to define this on arbitrary class functions of H by means of the explicit formula given in 4.10, but the trouble with this is that transitivity of induction is not entirely obvious. It is perhaps easier to observe that the characters of simple representations of H form a basis of Ccc(H) . We have defined G χ ↑G H on these basis elements, and we may define ↑H on arbitrary class functions so that it is a linear map. With these definitions the formulas of 4.14 hold for arbitrary class functions. We may also interpret Frobenius reciprocity within this framework. The inner products h , iG and h , iH provide us with the notion of the transpose of a linear map between the vector spaces Ccc(H) and Ccc(G) . Now Frobenius reciprocity states that induction and restriction are the transpose of each other. We know that the characters

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of simple modules form orthonormal bases of these vector spaces. Taking matrices with respect to these bases, the matrix of induction is the transpose of the matrix of restriction. (4.15) Examples. 1. Frobenius reciprocity is a most useful tool in calculating with induced characters. In the special case that V and W are simple representations over C of H and G, respectively, where H ≤ G, it says that the multiplicity of W as a summand of G V ↑G H equals the multiplicity of V as a summand of W ↓H . As an example we may take both V and W to be the trivial representations of their respective groups. As explained in 4.9, C ↑G H is a permutation module. We deduce from Frobenius reciprocity that as representations of G, C is a direct summand of C ↑G H with multiplicity one. n 2 −1 −1 2. Let G = hx, y x = y = 1, yxy = x i = D2n , the dihedral group of order 2n. Suppose that n is odd. We compute that the commutator [y, x] = xn−2 , and since n is odd we have G′ = hxn−2 i = hxi ∼ = Cn and G/G′ ∼ = C2 . Thus G has two complex characters of degree 1 which we denote 1 and −1. 2πi Let χs denote the degree 1 character of hxi specified by χs (xr ) = ζ rs where ζ = e n . Then χs ↑G hxi has values given in the following table.

1 −1 χs ↑G hxi

(1 ≤ s ≤

2n n n 1 x x2 1 1 1 1 1 1 2 ζ s + ζ s ζ 2s + ζ 2s

n

n−1 2

··· x ··· 1 ··· 1 n−1 n−1 ··· ζ 2 s +ζ 2 s

2 y 1 −1 0

n−1 2 )

TABLE: The character table of D2n , n odd. We verify that G hχs ↑G hxi , ±1iG = hχs , ±1 ↓hxi ihxi = 0

if n6 s, using Frobenius reciprocity (or a direct calculation), and hence the characters χs they are distinct, and so we have constructed are simple when n 6 s. For 1 ≤ s ≤ n−1 2 n−1 n+3 + 2 = 2 simple characters. This equals the number of conjugacy classes of G, so 2 we have the complete character table.

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Symmetric and Exterior Powers As further ways of constructing new representations from old ones we describe the symmetric powers and exterior powers of a representation. If V is a vector space over a field k its nth symmetric power is the vector space S n (V ) = V ⊗n /I where V ⊗n = V ⊗ · · · ⊗ V with n factors, and I is the subspace spanned by tensors (· · · ⊗ vi ⊗ · · · ⊗ vj ⊗ · · · − · · · ⊗ vj ⊗ · · · ⊗ vi ⊗ · · ·) where vi , vj ∈ V . We write the image of the tensor v1 ⊗ · · · ⊗ vn in S n (V ) as a (commutative) product v1 · · · vn , noting that in S n (V ) it does not matter in which order we write the terms. A good way to think of S n (V ) is as the space of homogeneous polynomials of degree n in a polynomial ring. Indeed, if u1 , . . . , ur is any basis of V and we let k[u1 , . . . , ur ]n denote the vector space of homogeneous polynomials of degree n in the ui as indeterminates, there is a surjective linear map V ⊗n → k[u1 , . . . , ur ]n ui1 ⊗ · · · ⊗ uin 7→ ui1 · · · uin (extended by linearity to the whole of V ⊗n ). This map contains I in its kernel, so there is induced a map S n (V ) → k[u1 , . . . , ur ]n . 1 2 r This is now an isomorphism since, modulo I, the tensors u⊗a ⊗ u⊗a ⊗ · · · ⊗ u⊗a r 1 2 Pr where i−1 ai = n, span V ⊗n , and they map to the monomials which form a basis of
n+r−1 dimk k[u1 , . . . , ur ]n . As is well-known, dimk k[u1 , . . . , ur ]n = . n The nth exterior power of V is the vector space

Λn (V ) = V ⊗n /J where J is the subspace spanned by tensors (· · ·⊗ vi ⊗ · · ·⊗ vj ⊗ · · ·+ · · ·⊗ vj ⊗ · · ·⊗ vi ⊗ · · ·) and (· · · ⊗ vi ⊗ · · · ⊗ vi ⊗ · · ·) where vi , vj ∈ V . We write the image of v1 ⊗ · · · ⊗ vn in Λn (V ) as v1 ∧ · · · ∧ vn , so that interchanging vi and vj changes the sign of the symbol, and if two of vi and vj are equal the symbol is zero. If the characteristic of k is not 2 the second of these properties follows from the first, but for the sake of characteristic 2 we impose it anyway. By an argument similar see that Λn (V ) to the one used for symmetric powers we
has as a basis {ui1 ∧ · · · ∧ uin 1 ≤ i1 , · · · in ≤ r}, and its dimension is nr . In particular, Λn (V ) = 0 if n > dim V . Suppose now that a group G acts on V and consider the diagonal action of G on V ⊗n . The subspaces of relations I and J are preserved by this action, and so there arise actions of G on S n (V ) and Λn (V ): g · (v1 v2 · · · vn ) = (gv1 )(gv2 ) · · · (gvn ) g · (v1 ∧ · · · ∧ vn ) = (gv1 ) ∧ · · · ∧ (gvn ).

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Because we substitute the expressions for gvi into the monomials which form the bases of S n (V ) and Λn (V ), we say that G acts on these spaces by linear substitutions. With these actions we have described the symmetric and exterior powers of the representation V . (4.16) Example. Consider the representation of G = hx x3 = 1i on the vector space V with basis {u1 , u2 } given by xu1 = u2 xu2 = −u1 − u2 . Then S 2 (V ) has a basis {u21 , u1 u2 , u22 } and x · u21 = u22 x · (u1 u2 ) = u2 (−u1 − u2 ) = −u1 u2 − u22 x · u22 = (−u1 − u2 )2 = u21 + 2u1 u2 + u22 . Similarly Λ2 (V ) has basis {u1 ∧ u2 } and x · (u1 ∧ u2 ) = u2 ∧ (−u1 − u2 ) = u1 ∧ u2 . The symmetric and exterior powers fit into a more general framework where we consider tensors with different symmetry properties. There is an action of the symmetric group Sn on the n-fold tensor power V ⊗n given by permuting the positions of vectors in a tensor, so that for example if α, β, γ are vectors in V then (1, 2)(α ⊗ β ⊗ γ) = β ⊗ α ⊗ γ, (1, 3)(β ⊗ α ⊗ γ) = γ ⊗ α ⊗ β. From the above very convincing formulas and the fact that (1, 2, 3) = (1, 3)(1, 2) we deduce that (1, 2, 3)(α ⊗ β ⊗ γ) = γ ⊗ α ⊗ β which is evidence that if σ ∈ Sn then σ(v1 ⊗ · · · ⊗ vn ) = vσ−1 (1) ⊗ vσ−1 (2) ⊗ · · · ⊗ vσ−1 (n) , a formula which is not quite so obvious. With this action it is evident that S n (V ) is the largest quotient of V ⊗n on which Sn acts trivially, and when char(k) 6= 2, Λn (V ) is the largest quotient of V ⊗n on which Sn acts as a sum of copies of the sign representation. We define the symmetric tensors to be the fixed points (V ⊗n )Sn , and when char k 6= 2 we define the skew-symmetric tensors to be the largest kSn -submodule of V ⊗n which is a sum of modules isomorphic to the sign representation. Thus symmetric tensors = {w ∈ V ⊗n σ(w) = w for all σ ∈ Sn }, skew-symmetric tensors = {w ∈ V ⊗n σ(w) = sign(σ)w for all σ ∈ Sn }.

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When we let G act diagonally on V ⊗n the symmetric tensors, the skew-symmetric tensors, as well as the subspaces I and J defined earlier remain invariant for the action of G. We easily see this directly, but at a more theoretical level the reason is that the actions of G and Sn on V ⊗n commute with each other (as is easily verified), so that V ⊗n acquires the structure of a k[G × Sn ]-module, and elements of G act as endomorphisms of V ⊗n as a kSn -module, and vice-versa. Every endomorphism of the kSn -module V ⊗n must send the Sn -fixed points to themselves, for example, and so the symmetric tensors are invariant under the action of G. One sees similarly that the other subspaces are also invariant under the action of G. We remark that, in general, the symmetric power S n (V ) and the symmetric tensors provide non-isomorphic representations of G, as do Λn (V ) and the skew-symmetric tensors. This phenomenon is investigated in the exercises at the end of this section. However these pairs of kG-modules are isomorphic in characteristic zero, and we now consider in detail the case of the symmetric and exterior square. Suppose that k is a field whose characteristic is not 2. In this situation the only tensor which is both symmetric and skew-symmetric is 0, and furthermore any tensor may be written as the sum of a symmetric tensor and a skew-symmetric tensor in the following way: X

λij vi ⊗ vj =

1X 1X λij (vi ⊗ vj + vj ⊗ vi ) + λij (vi ⊗ vj − vj ⊗ vi ). 2 2

We deduce from this that V ⊗ V = symmetric tensors ⊕ skew-symmetric tensors as kG-modules. The subspace I which appeared in the definition S 2 (V ) = (V ⊗ V )/I is contained in the space of skew-symmetric tensors, and the subspace J for which Λ2 (V ) = (V ⊗ V )/J is contained in the space of symmetric tensors. By counting dimensions we see that dim I + dim J = dim V ⊗ V and putting this together we see that I = skew-symmetric tensors J = symmetric tensors, and V ⊗ V = I ⊕ J. From this information we see on factoring out I and J that S 2 (V ) ∼ = symmetric tensors Λ2 (V ) ∼ = skew-symmetric tensors and we have proved the following result.

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(4.17) PROPOSITION. Suppose V is a representation for G over a field k whose characteristic is not 2. Then V ⊗V ∼ = S 2 (V ) ⊕ Λ2 (V ), as kG-modules, where G acts diagonally on V ⊗ V and by linear substitutions on S 2 (V ) and Λ2 (V ). Our application of this is that when constructing new representations from an existing representation V , the tensor square will always decompose in this fashion. Suppose now that k = C. If χ is the character of a representation V we write S 2 χ and Λ2 χ for the characters of S 2 (V ) and Λ2 (V ). (4.18) PROPOSITION. Let χ be the character of a representation V of G over C. Then 1 S 2 χ(g) = (χ(g)2 + χ(g 2 )) 2 1 2 Λ χ(g) = (χ(g)2 − χ(g 2 )). 2 Proof. For each g ∈ G, V ↓G hgi is the direct sum of 1-dimensional representations of the cyclic group hgi, and so we may choose a basis u1 , . . . , ur for V such that g · ui = λi ui for scalars λi . The monomials u2i with 1 ≤ i ≤ r and ui uj with 1 ≤ i < j ≤ r form a basis for S 2 V , and so the eigenvalues of g on this space are λ2i with 1 ≤ i ≤ r and λi λj with 1 ≤ i < j ≤ r. Therefore 2

S χ(g) =

r X

λ2i +

i=1

X

λi λj

1≤i<j≤r

1 ((λ1 + · · · + λr )2 + (λ21 + · · · + λ2r )) 2 1 = (χ(g)2 + χ(g 2 )). 2

=

Similarly Λ2 V has a basis ui ∧ uj with 1 ≤ i < j ≤ r, so the eigenvalues of g on Λ2 V are λi λj with 1 ≤ i < j ≤ r and Λ2 χ(g) =

X

λi λj

1≤i<j≤r

1 ((λ1 + · · · + λr )2 − (λ21 + · · · + λ2r )) 2 1 = (χ(g)2 − χ(g 2 )). 2

=

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The Construction of Character Tables We may now summarize some major techniques used in constructing complex character tables. The first things to do are to determine the conjugacy classes in G, the abelianization G/G′ , the 1-dimensional characters of G. We construct characters of degree larger than 1 as natural representations of G, representations induced from subgroups, tensor products of other representations, symmetric and exterior powers of other representations, contragredients of other representations. As a special case of the induced representations, we have permutation representations. The representations obtained by these methods might not be simple, so we test them for simplicity and subtract off known character summands using the orthogonality relations which are assisted in the case of induced characters, by Frobenius reciprocity. The orthogonality relations provide a check on the accuracy of our calculations, and also enable us to complete the final row of the character table. The facts that the character degrees divide |G| and that the sum of the squares of the degrees equals |G| also help in this. Exercises for Section 4. 1. Compute the character table of the dihedral group D2n when n is even. 2. Let G be the non-abelian group of order 21: G = hx, y x7 = y 3 = 1, yxy −1 = x2 i.

Show that G has 5 conjugacy classes, and find its character table. 3. Find the character table of the following group of order 36: G = ha, b, c a3 = b3 = c4 = 1, ab = ba, cac−1 = b, cbc−1 = a2 i.

[It follows from these relations that ha, bi is a normal subgroup of G of order 9.]

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4. Given any representation ρ : G → GL(V ) where V is a vector space over any field k, evidently Ker ρ is a normal subgroup of G. Prove the following ‘converse’ to the last statement, namely: given a normal subgroup N ⊳ G, there exists a representation ρ with Ker ρ = N . If we assume further that k = C, show how to identify Ker ρ knowing only the character of ρ. 5. Let k be any field, H a subgroup of G, and V a representation of H over k. Show G ∗ ∗ ∼ ∼ that V ∗ ↑G H = (V ↑H ) . Deduce that kG = (kG) and (more generally) that permutation modules are self-dual (i.e. isomorphic to their dual). 6. Let k be any field, and V any representation of G over k. Prove that V ⊗ kG is isomorphic to a direct sum of copies of kG. 7. Three-suffix tensors have components τijk ∈ R where i, j, k ∈ {1, 2, 3}, and form a vector space V of dimension 27 over R. The symmetric group S3 acts on V by permuting the suffixes. Decompose the space V as a direct sum of simple representations of S3 , giving the mutiplicities of each simple representation. [Observe that V is a permutation representation.] Give also the decomposition of V as a direct sum of three subspaces consisting of tensors with different symmetry properties under S3 . What are the dimensions of these subspaces? 8. Let V be a representation of G over a field k of characteristic zero. Prove that the symmetric power S n (V ) is isomorphic as a kG-module to the space of symmetric tensors in V ⊗n . 9. Let U, V be kG-modules where k is a field, and suppose we are given a nondegenerate bilinear pairing h , i:U ×V →k which has the property hu, vi = hgu, gvi for all u ∈ U , v ∈ V , g ∈ G. If U1 is a subspace hu, vi = 0 for all u ∈ U1 } and if V1 is a subspace of V let of U let U1⊥ = {v ∈ V ⊥ V1 = {u ∈ U hu, vi = 0 for all v ∈ V1 }. (a) Show that V ∼ = U ∗ as kG-modules, and that there is an identification of V with U ∗ so that h , i identifies with the canonical pairing U × U ∗ → k. (b) Show that if U1 and V1 are kG-submodules, then so are U1⊥ and V1⊥ . (c) Show that if U1 ⊆ U2 are kG-submodules of U then U1⊥ /U2⊥ ∼ = (U2 /U1 )∗ as kG-modules. (d) Show that the composition factors of U ∗ are the duals of the composition factors of U . 10. Let Ω be a G-set and kΩ the corresponding permutation module, where k is a field. Let h , i : kΩ × kΩ → k be the symmetric bilinear form specified on the elements of Ω as n 1 if ω1 = ω2 , hω1 , ω2 i = 0 otherwise.

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(a) Show that this bilinear form is G-invariant, i.e. hω1 , ω2 i = hgω1 , gω2 i for all g ∈ G. (b) Show that kΩ is self-dual, i.e. kΩ ∼ = (kΩ)∗ . 11. Let V be a kG-module where k is a field, and let h , i : V × V ∗ → k be the canonical pairing between V and its dual, so hv, f i = f (v). (a) Show that the specification hv1 ⊗ · · · ⊗ vn , f1 ⊗ · · · ⊗ fn i = f1 (v1 ) · · · fn (vn ) determines a non-degenerate bilinear pairing h , i : V ⊗n × (V ∗ )⊗n → k which is invariant both for the diagonal action of G and the action of Sn given by permuting the positions of the tensors. (b) Let I and J be the subspaces of V ⊗n which appear in the definitions of the symmetric and exterior powers, so S n (V ) = V ⊗n /I and Λ⊗n = V ⊗n /J. Show that I ⊥ equals the space of symmetric tensors in (V ∗ )⊗n , and that J ⊥ equals the space of skewsymmetric tensors in (V ∗ )⊗n (at least, when char k 6= 2). (c) Show that (S n (V ))∗ ∼ = STn (V ∗ ), and that (Λn (V ))∗ ∼ = SSTn (V ∗ ), where STn denotes the symmetric tensors, and in general we define the skew-symmetric tensors SSTn (V ∗ ) to be J ⊥ . 12. Let G = C2 × C2 be the Klein four group with generators a and b, and k = F2 the field of two elements. Let V be a 3-dimensional space on which a and b act via the matrices     1 0 0 1 0 0  1 1 0  and  0 1 0  . 0 0 1 1 0 1 Show that S 2 (V ) is not isomorphic to either ST 2 (V ) or ST 2 (V )∗ , where ST denotes the symmetric tensors. [Hint: Compute the dimensions of the spaces of fixed points of these representations.] 13. (Artin’s Induction Theorem) Let Ccc(G) denote the vector space of class functions on G and let C be a set of subgroups of G which contains a representative of each conjugacy class of cyclic subgroups of G. Consider the linear mappings resC : Ccc(G) → and indC :

M

M

Ccc(H)

H∈C

Ccc(H) → Ccc(G)

H∈C

whose component homomorphisms are the linear mappings given by restriction cc(G) ↓G → Ccc(H) H: C

and induction cc(H) ↑G → Ccc(G) H: C

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(a) With respect to the usual inner product h , iG on Ccc(G) and the inner product L on H∈C Ccc(H) which is the orthogonal sum of the h , iH , show that resC and indC are the transpose of each other. (b) Show that resC is injective. [Use the fact that Ccc(G) has a basis consisting of characters, which take their information from cyclic subgroups.] (c) Prove Artin’s induction theorem: In Ccc(G) every character χ can be written as a rational linear combination X χ= aH,ψ ψ ↑G H

where the sum is taken over cyclic subgroups H of G, ψ ranges over characters of H and aH,ψ ∈ Q. [Deduce this from surjectivity of indC and the fact that it is given by a matrix with integer entries. A stronger statement of Artin’s theorem is possible: there is a proof due to Brauer which gives an explicit formula for the coefficients aH,ψ ; from this we may deduce that when χ is the character of a QG-module the ψ which arise may all be taken to be the trivial character.] (d) Show that if U is any CG-module then there are CG-modules P and Q, each a direct sum of modules of the form V ↑G H where H is cyclic, for various V and H, so that n Q for some n, where U is the direct sum of n copies of U . Un ⊕ P ∼ = 14. (Molien’s Theorem) (a) Let ρ : G → GL(V ) be a complex representation of G, so that V is a CG-module, and for each n let χS n (V ) be the character of the nth symmetric power of V . Show that for each g ∈ G there is an equality of formal power series ∞ X

χS n (V ) (g)tn =

n=0

1 . det(1 − tρ(g))

Here t is an indeterminate, and the determinant which appears in this expression is of a matrix with entries in the polynomial ring C[t], so that the determinant is a polynomial in t. On expanding the rational function on the right we obtain a formal power series which is asserted to be equal to the formal power series on the left. [Choose a basis for V so that g acts diagonally, with eigenvalues ξ1 , . . . , ξd . Show that on P both sides of the equation the coefficient of tn is equal to i1 +···+id =n ξ1i1 · · · ξdid .] (b) If W is a simple CG-module we may write the multiplicity of W as a summand n of S (V ) as hχS n (V ) , χW i and consider the formal power series MV (W ) =

∞ X

hχS n (V ) , χW itn .

i=0

Show that MV (W ) =

1 X χW (g −1 ) . |G| det(1 − tρ(g)) g∈G

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(c) When G = S3 and V is the 2-dimensional simple CS3 -module show that MV (C) =

1 (1 −

t2 )(1

− t3 )

= 1 + t2 + t3 + t4 + t5 + 2t6 + t7 + 2t8 + 2t9 + 2t10 + · · ·

t3 = t3 + t5 + t6 + t7 + t8 + 2t9 + t10 + · · · MV (ǫ) = 2 3 (1 − t )(1 − t ) t(1 + t) MV (V ) = = t + t2 + t3 + 2t4 + 2t5 + 2t6 + 3t7 + 3t8 + 3t9 + 4t10 + · · · (1 − t2 )(1 − t3 ) where C denotes the trivial module and ǫ the sign representation. Deduce, for example, that the eighth symmetric power S 8 (V ) ∼ = C2 ⊕ ǫ ⊕ V 3 .

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5. More on Induction and Restriction: Theorems of Mackey and Clifford We start with Mackey’s decomposition formula, which is a further relationship between induction and restriction. For this we need to consider double cosets. Given subgroups H and K of G we define for each g ∈ G the (H, K)-double coset HgK = {hgk h ∈ H, k ∈ K}.

If Ω is a left G-set we use the notation G\Ω for the set of orbits of G on Ω, and denote a set of representatives for the orbits by [G\Ω]. Similarly if Ω is a right G-set we write Ω/G and [Ω/G]. We will use all the time the fact that if Ω is a transitive G-set and ω ∈ Ω then Ω∼ = G/ StabG (ω), the set of left cosets of the stabilizer of ω in G. (5.1) PROPOSITION. Let H, K ≤ G. (1) Each (H, K)-double coset is a disjoint union of right cosets of H and a disjoint union of left cosets of K. (2) Any two (H, K)-double cosets either coincide or are disjoint. The (H, K)-double cosets partition G. (3) The set of (H, K)-double cosets is in bijection with the orbits H\(G/K), and with the orbits (H\G)/K under the mappings HgK 7→ H(gK) ∈ H\(G/K) HgK 7→ (Hg)K ∈ (H\G)/K. Proof. (1) If hgk ∈ HgK and k1 ∈ K then hgk · k1 = hg(kk1 ) ∈ HgK so that the entire left coset of K which contains hgk is contained in HgK. This shows that HgK is a union of left cosets of K, and similarly it is a union of right cosets of K. −1 (2) If h1 g1 k1 = h2 g2 k2 ∈ Hg1 K ∩ Hg2 K then g1 = h−1 ∈ Hg2 K so that 1 h2 g2 k2 k1 Hg1 K ⊆ Hg2 K, and similarly Hg2 K ⊆ Hg1 K. Thus if two double cosets are not disjoint, they coincide. (3) In the statement of this result, G acts from the left on the left cosets G/K, hence so does H by restriction of the action. We denote the set of H-orbits on G/K by H\(G/K). The mapping {double cosets} → H\(G/K) HgK 7→ H(gK) is evidently well-defined and surjective. If H(g1 K) = H(g2 K) then g2 K = hg1 K for some h ∈ H, so g2 ∈ Hg1 K and Hg1 K = Hg2 K by (2). Hence the mapping is injective. The proof that double cosets biject with (H\G)/K is similar.

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In view of (3) we denote the set of (H, K)-double cosets in G by H\G/K. We denote a set of representatives for these double cosets by [H\G/K]. (5.2) Example. Consider S2 = {1, (12)} as a subgroup of S3 . We have S2 \S3 /S2 = {{1, (12)}, {(123), (132), (13), (23)}}, while, for example, [S2 \S3 /S2 ] = {1, (123)}. S3 acts transitively on {1, 2, 3} with StabS3 (3) = S2 , so as S3 -sets we have S3 /S2 ∼ = {1, 2, 3}. Thus the set of orbits on this set under the action of S2 is S2 \(S3 /S2 ) ↔ {{1, 2}, {3}}. We observe that these orbits are indeed in bijection with the double cosets S2 \S3 /S2 . This example illustrates the point that when computing double cosets it may be advantageous to identify G/K as some naturally occuring G-set, rather than as the set of left cosets. In the next result we distinguish between conjugation on the left and on the right: g x = gxg −1 and xg = g −1 xg. Later on we will write cg (x) = gx, so that cg : H → gH is the homomorphism which is left conjugation by g, and cg −1 (x) = xg . (5.3) PROPOSITION. Let H, K be subgroups of G and g ∈ G an element. We have isomorphisms HgK/K ∼ = H/(H ∩ gK) as left H-sets and H\HgK ∼ = (H g ∩ K)\K

as right K-sets.

Thus the double coset HgK is a union of |H : H ∩ gK| left K-cosets and |K : H g ∩ K| right H-cosets. We have |G : K| =

X

|H : H ∩ gK|

X

|K : H g ∩ K|.

g∈[H\G/K]

and |G : H| =

g∈[H\G/K]

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Proof. HgK is the union of a single H-orbit of left K-cosets. The stablizer in H of one of these is StabH (gK) = {h ∈ H hgK = gK} = {h ∈ H hg K = K} = {h ∈ H hg ∈ K} = H ∩ gK.

Thus HgK/K ∼ = H/(H ∩ gK) as left H-sets and the number of left K-cosets in HgK equals |H : H ∩ gK|. By summing these numbers over all double cosets we obtain the total number of left K-cosets |G : K|. The argument with right H-cosets is similar. We next introduce conjugation of representations, a concept we have in fact already met with induced representations. Suppose H is a subgroup of G, g ∈ G and V is a representation of H. We define a representation gV of gH by specifying that gV = V as a set, and if gh ∈ gH then gh·v = hv. Thus if ρ : H → GL(V ) was the original representation, cg−1

ρ

the conjugate representation is the composite homomorphism gH −→H −→GL(V ) where cg −1 ( gh) = h. L When studying the structure of induced representations V ↑G H= g∈[G/H] g ⊗ V , the g subspace g ⊗ V is in fact a representation for H; for ghg −1 · (g ⊗ v) = ghg −1 g ⊗ v = gh ⊗ v = g ⊗ hv. When g ⊗ V is identified with V via the linear isomorphism g ⊗ v 7→ v the action of gH on V which arises coincides with the action we have just described on gV . (5.4) THEOREM (Mackey decomposition formula). Let H, K be subgroups of G and V a representation for K over a commutative ring R. Then M (V ↑G ) ↓G ∼ ( g(V ↓K g )) ↑H g = K

H

g∈[H\G/K]

H ∩K

H∩ K

as RH-modules. Proof. We have V ↑G K= The terms

L

x∈[G/K] x

⊗ V . Consider a particular double coset HgK.

M

x⊗V

x∈[G/K] x∈HgK

form an R-submodule invariant under the action of H, since it is the direct sum of an orbit of R-submodules permuted by H. Now StabH (g ⊗ V ) = {h ∈ H hg ∈ gK} = H ∩ gK.

Therefore as a representation for H this subspace is (g ⊗ V ) ↑H H∩ gK . As observed before g K ∼ the statement of this theorem we have g ⊗ V = (V ↓H g ∩K ) as a representation of H ∩ gK. Putting these expressions together gives the result.

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As an application of Mackey’s theorem we consider permutation modules arising from multiply transitive G-sets. We say that a G-set Ω is n-transitive if for every pair of ntuples (a1 , . . . , an ) and (b1 , . . . , bn ) in which the ai are distinct elements of Ω and the bi are distinct elements of Ω, there exists g ∈ G with gai = bi for every i. For example, Sn acts n-transitively on {1, . . . , n}, and one may show that An acts (n − 2)-transitively on {1, . . . , n} provided n ≥ 3. Notice that if G acts n-transitively on Ω then it also acts (n − 1)-transitively. (5.5) LEMMA. Let Ω be a G-set. Then G acts n-transitively on Ω if and only if G acts transitively on Ω and for any ω ∈ Ω, StabG (ω) acts (n − 1)-transitively on Ω − {ω}. Proof. If G acts n-transitively then G also acts transitively, and if a2 , . . . , an and b2 , . . . , bn are two sets of n − 1 distinct points of Ω, none of them equal to ω, then there exists g ∈ G so that g(ω) = (ω) and g(ai ) = bi for all i. This shows that StabG (ω) acts (n − 1)-transitively on Ω − {ω}. Conversely, suppose G acts transitively on Ω and StabG (ω) always acts (n − 1)transitively on Ω − {ω}. Let a1 , . . . , an and b1 , . . . , bn be two sets of n distinct points of Ω. We may find g1 ∈ G so that g1 a1 = b1 . Now find g2 ∈ StabG (b1 ) so that g2 g1 ai 2 = bi for 2 ≤ i ≤ n and put g = g2 g1 . Then g(ai ) = (bi ) for all i. (5.6) PROPOSITION. Whenever Ω is a G-set the permutation module CΩ may be written as a direct sum of CG-modules CΩ = C ⊕ V Suppose that |Ω| ≥ 1, so V 6= 0. The representation V is simple if and only if G acts 2-transitively on Ω. In that case, V is not the trivial representation. Proof. Pick any orbit of G on Ω. It is isomorphic as a G-set to G/H for some subgroup H ≤ G and so C[G/H] is a direct summand of CΩ, with character 1 ↑G H . Since h1, 1 ↑G H iG = h1, 1iH = 1 we deduce that C is a summand of C[G/H] and hence of CΩ. In the equivalence of statements which forms the third sentence, neither side is true if G has more than one orbit on Ω, so we may assume Ω = G/H. The character of CΩ is

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1 ↑G H , and we compute G G G h1 ↑G H , 1 ↑H iG = h(1 ↑H ) ↓H , 1iH X =h ( g1) ↑H H∩ gH , 1iH g∈[H\G/H]

=

X

h1 ↑H H∩ gH , 1iH

g∈[H\G/H]

=

X

h1, 1iH∩ gH

g∈[H\G/H]

=

X

1

g∈[H\G/H]

= |H\G/H|, using Frobenius reciprocity twice and Mackey’s formula. Now |H\G/H| is the number of orbits of H (the stabilizer of a point) on G/H. By Lemma 5.5 this number is 2 if G acts 2-transitively on Ω, and otherwise it is greater than 2 (since |Ω| ≥ 1). Writing G C[G/H] = S1 ⊕· · ·⊕Sn as a direct sum of simple representations we have h1 ↑G H , 1 ↑H iG ≥ n and we get the value 2 for the inner product if and only if there are 2 simple representations in this expression, and they are non-isomorphic. One of S1 and S2 must be C and the other V , so V is not the trivial representation. (5.7) Example. Let Ω = {1, . . . , n} acted upon by Sn and also An . Then CΩ ∼ = C⊕V where V is a simple representation of Sn , which remains simple on restriction to An provided n ≥ 4. We now turn to Clifford’s theorem, which we present in a weak and a strong form. The weak form is used as a step in proving the strong form a little later, and as a result in its own right it only has force in a situation where |G| is not invertible in the ground ring. (5.8) THEOREM (Weak form of Clifford’s theorem). Let k be any field, U a simple kG-module and N a normal subgroup of G. Then U ↓G N is semisimple as a kN -module. Proof. Let V be any simple kN -submodule of U . For every g ∈ G, gV is also a kN -submodule since if n ∈ N we have n(gv) = g(g −1 ng)v ∈ gV , using the fact that N is normal. Evidently gV is also simple, since if W were a kN -submodule of gV then g −1 W P would be a submodule of V . Now g∈G gV is a non-zero G-invariant subspace of the P simple kG-module U , and so g∈G gV = U . As a kN -module we see that U ↓G N is the sum of simple submodules, and hence U ↓G N is semisimple by the results of Section 1.

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The kN -submodules gV which appear in the proof of 5.8 are isomorphic to modules we have seen before. Since N ⊳G, the conjugate module gV is a representation for gN = N . The mapping g V → gV v 7→ gv is an isomorphism of kN -modules, since if n ∈ N the action on gV is n · v = g −1 ngv and the action on gV is n(gv) = g(g −1 ngv). Recall also that these modules appeared when we described induced modules. Part of Clifford’s theorem states that the simple module U is in fact an induced module. (5.9) THEOREM (Clifford’s theorem). Let k be any field, U a simple kG-module and a1 ar N a normal subgroup of G. Write U ↓G N = S1 ⊕ · · · ⊕ Sr where the Si are non-isomorphic simple kN -modules and the ai are the multiplicities to which they occur. (We refer to the summands Siai as the homogeneous components.) Then (1) G permutes the homogeneous components transitively, (2) a1 = a2 = · · · = ar , and (3) if H = StabG (S1a1 ) then U ∼ = S1a1 ↑G H. Proof. We start by observing that the homogeneous component Siai is characterized as the unique largest kN -submodule which is isomorphic to a direct sum of copies of Si , as stated in Corollary 1.9. If g ∈ G then g(Siai ) is a direct sum of isomorphic simple modules gSi , and so by this characterization must be contained in another homogeneous a component: g(Siai ) ⊆ Sj j for some j. Since U = g(S1a1 ) ⊕ · · · ⊕ g(Srar ), by counting a dimensions we have g(Siai ) = Sj j . Thus G permutes the homogeneous components. Since P a1 g∈G g(S1 ) is a non-zero G-invariant submodule of the simple module U , it must equal U , and so the action on the homogeneous components is transitive. This establishes (1), a and (2) follows since for any pair (i, j) we can find g ∈ G with g(Si )ai = Sj j , so ai = aj . Finally, (3) is a direct consequence of Proposition 4.8. For now, we give just one application of Clifford’s theorem; we will see more when we come to consider representations of p-groups in characteristic p. Although true as stated, the next corollary only has force when k is a field of characteristic zero, as we will see in the next section that the only simple representation for a p-group in characteristic p is the trivial representation. (5.10) COROLLARY. Let k be any algebraically closed field and G a p-group. Then every simple module for G has the form U ↑G H where U is a 1-dimensional module for some subgroup H. Proof. We proceed by induction on |G|. Let ρ : G → GL(S) be a simple representation of G over k and put N = Ker ρ. Then S is really a representation of G/N . If N 6= 1 then

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G/N is a group of smaller order than G, so by induction S has the claimed structure as a representation of G/N , and hence also as a representation of G. Thus we may assume N = 1 and G embeds in GL(S). If G is abelian then all simple representations are 1-dimensional, so we are done. Assume now that G is not abelian. Then G has a normal abelian subgroup A which is not central. To construct this subgroup A, let Z2 (G) denote the second centre of G, that is, the preimage in G of Z(G/Z(G)). If x is any element of Z2 (G) − Z(G) then A = hZ(G), xi is a normal abelian subgroup not contained in Z(G). We apply Clifford’s theorem: a1 ar S ↓G A = S1 ⊕ · · · ⊕ Sr a1 a1 and S = V ↑G K where V = S1 and K = StabG (S1 ). We argue that V must be a simple kK-module, since if it had a proper submodule W then W ↑G K would be a proper submodule of S, which is simple. If K 6= G then by induction V = U ↑K H where U is K G G 1-dimensional, and so S = (U ↑H ) ↑K = U ↑H has the required form. We show finally that the case K = G cannot happen. For if it were to happen then G S ↓A = S1a1 and since A is abelian dim S1 = 1. The elements of A must therefore act via scalar multiplication on S. Since such action would commute with the action of G, which is faithfully represented on S, we deduce that A ⊆ Z(G), a contradiction.

The conclusion of Corollary 5.10 also applies to supersoluble groups, which also have the property, if they are not abelian, that they have a non-central normal abelian subgroup. Exercises for Section 5. 1. Let k be any field, and g any element of a finite group G. (a) If K ≤ H ≤ G are subgroups of G, V a kH-module, and W a kK-module, show g g H H∼ g H g ∼ g that (( gV ) ↓ gH K = (V ↓K ) and (( W ) ↑ gK = (W ↑K ). [This allows one to interchange conjugation with induction, or with restriction, in Mackey’s formula.] (b) If U is any kG-module, show that gU ∼ = U. 2. Consider the complex representations of Sn and An . Show that if S is any simple representation of Sn then S ↓An is the sum of at most 2 simple representations of An , and that if the degree of S is odd then S ↓An is simple. In the situation where S ↓An is the sum of 2 simple representations of An , show that S is induced from a representation of An . In the situation where S ↓An is simple, show that S ↓An↑Sn ∼ = S ⊕ (ǫ ⊗ S) where ǫ is the ∼ sign representation, and that S 6= ǫ ⊗ S.

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6. Representations of p-groups in characteristic p The study of representations of a group over a field whose characteristic divides the group order is more delicate than the case of ordinary representation theory (characteristic zero), since modules need not be semisimple, and we have to do more than count multiplicities of simple modules. The notion of a direct sum decomposition is still the first consideration in describing the structure of a representation, but now we may find ourselves in the situation where we have broken apart the modules as far as possible into direct summands, and still these summands are not simple. An example of this was given in 1.2.2. In view of this phenomenon we make the definition that a module U for an algebra A is indecomposable if it cannot be expressed as a direct sum of two modules except in a trivial way, that is if U ∼ = V ⊕ W then either V = 0 or W = 0. We start with an explicit description of the representations of cyclic p-groups. n (6.1) PROPOSITION. Let k be any field of characteristic p and let G = hg g p = 1i. n Then there is a ring isomorphism kG ∼ = k[X]/(X p ), where k[X] is the polynomial ring in an indeterminate X. Proof. We define a mapping n

G → k[X]/(X p ) g s 7→ (X + 1)s . Since n

n

n

(X + 1)p = X p + p(· · ·) + 1 ≡ 1 (mod(X p )) this mapping is a group homomorphism, and hence it extends to a linear map n

kG → k[X]/(X p ) which is an algebra homomorphism. Since g s is sent to X s plus terms of lower degree, n n the images of 1, . . . , g p −1 form a basis of k[X]/(X p ). The mapping therefore gives a n bijection between a basis of kG and a basis of k[X]/(X p ), and so is an isomorphism. A module over a ring is said to be cyclic if it can be generated by one element. We now exploit the structure theorem for finitely-generated modules over a principal ideal domain, which says that such modules are direct sums of cyclic modules.

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(6.2) THEOREM. Let k be any field of characteristic p. Every finitely-generated n k[X]/(X p )-module is a direct sum of cyclic modules Ur = k[X]/(X r ) where 1 ≤ r ≤ pn . The only simple module is the 1-dimensional module U1 . Each module Ur has a unique composition series, and hence is indecomposable. n

Proof. The modules for k[X]/(X p ) may be identified with the modules for k[X] on n which X p acts as zero. Every finitely-generated k[X]-module is a direct sum of modules n k[X]/I where I is an ideal. Hence every k[X]/(X p )-module is a direct sum of modules n n k[X]/I on which X p acts as zero, which is to say (X p ) ⊆ I. The ideals I which satisfy n this last condition are the ideals (a) where a X p . This forces I = (X r ) where 1 ≤ r ≤ pn , and k[X]/I = Ur . The submodules of Ur must have the form J/(X r ) where J is some ideal containing (X r ), and they are precisely the submodules in the chain 0 ⊂ (X r−1 )/(X r ) ⊂ (X r−2 )/(X r ) ⊂ · · · ⊂ (X)/(X r ) ⊂ Ur . This is a composition series, since each successive quotient has dimension 1, and since it is a complete list of submodules, it is the only one. If we could write Ur = V ⊕ W as a non-trivial direct sum, then Ur would have at least 2 composition series, obtained by taking first a composition series for V , then one for W , or vice-versa. Hence each Ur is indecomposable and we have a complete list of the indecomposable modules. The only Ur which is simple is U1 , which is the trivial module. A module with a unique composition series is said to be uniserial. We see from the n description of k[X]/(X p )-modules that Ur has a basis 1+(X r ), X +(X r ), . . . , X r−1 +(X r ) so that X acts on Ur with matrix 

0 1  

0 ..

.

..

 .

1

0

 . 

Translating now to modules for kG where G is a cyclic p-group, the generator g acts on Ur as X + 1, which has matrix 

1 1  

1 ..

.

.. 1

 . 1

 . 

Thus we see that the indecomposable kG-modules are exactly given by specifying that the generator g acts via a matrix which is a single Jordan block, of size up to pn . It is often

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helpful to picture Ur as the diagram •  X=g−1 y •  X=g−1 y Ur = . .. •  X=g−1 y • One way to interpret this is that the vertices are in bijection with a basis of Ur , and the action of X or g − 1 is given by the arrows. Where no arrow is shown starting from a particular vertex (as happens in this case only with the bottom one), the interpretation is that X and g − 1 act as zero. (6.3) PROPOSITION. Let k be any field of characteristic p and G a p-group. The only simple kG-module is the trivial module. Proof. We offer two proofs of this. Proof 1. We proceed by induction on |G|, the induction starting when G is the identity group, for which the result is true. Suppose G 6= 1 and the result is true for p-groups of smaller order. There exists a normal subgroup N of G of index p. If S is any simple kG-module then by Clifford’s theorem S ↓G N is semisimple. By induction, N acts trivially on S. Thus S is really a representation of G/N which is cyclic of order p. We have just proved that the only simple representation of this group is the trivial representation. Proof 2. Let S be any simple kG-module and let 0 6= x ∈ S. The subgroup of S generated by the elements {gx g ∈ G} is invariant under the action of G, it is abelian and of exponent p, since it is a subgroup of a vector space in characteristic p. Thus it is a finite p group acted on by G. Consider the orbits of G on this finite group. Since G is a p-group the orbits have the form G/H where H is a subgroup, and so have size a power of p (or 1). The zero element is fixed by G, and we deduce that there must be another element fixed by G since otherwise the other orbits would all have size pn with n ≥ 1, and their union would not be a p-group. Thus there exists y ∈ S fixed by G, and now hyi is a trivial submodule of S. By simplicity it must equal S. As an application of this we can give some information about the simple representations of arbitrary finite groups in characteristic p. For this we observe that in every finite group G there is a unique largest normal p-subgroup of G, denoted Op (G). For if H and K are normal p-subgroups of G then so is HK, and thus the subgroup generated by all normal p-subgroups of G is a again a normal p-subgroup, which evidently contains all the others.

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(6.4) COROLLARY. Let k be any field of characteristic p, G any finite group and S a simple kG-module. Then Op (G) acts trivially on S. Thus the simple kG-modules are precisely the simple k[G/Op (G)]-modules, made into kG-modules via the homomorphism G → G/Op (G). Proof. By Clifford’s theorem, S ↓G Op (G) is semisimple. Therefore, by 6.3, Op (G) acts trivially on S. (6.5) Example. Let k be a field of characteristic 2, and consider the representations of A4 over k. Since O2 (A4 ) = C2 × C2 , the simple kA4 modules are the simple C3 = A4 /O2 (A4 )-representations, made into representations of A4 . Now kC3 is semisimple, and if k contains a primitive cube root of unity ω (i.e. if F4 ⊆ k) there are three 1-dimensional simple representations, on which the generator of C3 acts as 1, ω or ω 2 . At this point we examine further the structure of representations which are not semisimple, and we work in the context of modules for a ring A. If U is an A-module we defined the socle of U to be the sum of all the simple submodules of U , and we showed (at least in the case that U is finite-dimensional) that it is the unique largest semisimple submodule of U . We now work with quotients and define a dual concept, the radical of U . We work with quotients instead of submodules, and use the fact that if M is a submodule of U , the quotient U/M is simple if and only if M is a maximal submodule of U . We put \ Rad U = {M M ⊂ U is a maximal submodule}. In our applications U will always be Noetherian, so this intersection will be non-empty.

(6.6) LEMMA. Let U be a module for a ring A. (1) Suppose that M1 , . . . , Mn are maximal submodules of U . Then there is a subset I ⊆ {1, . . . , n} such that M U/(M1 ∩ · · · ∩ Mn ) ∼ U/Mi = i∈I

which, in particular, is a semisimple module. (2) Suppose further that U has the descending chain condition on submodules. Then U/ Rad U is a semisimple module, and Rad U is the unique smallest submodule of U with this property. Proof. (1) Let I be a subset of {1, . . . , n} maximal with the property that the T T quotient homomorphisms U/( i∈I Mi ) → U/Mi induce an isomorphism U/( i∈I Mi ) ∼ = L T · · · ∩ Mn and argue by contradiction. If it i∈I U/Mi . We show that i∈I Mi = M1 ∩ T were not the case, there would exist Mj with i∈I Mi 6⊆ Mj . Consider the homomorphism M f :U →( U/Mi ) ⊕ U/Mj i∈I

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whose components are the quotient homomorphisms U → U/Mk . This has kernel Mj ∩ T i∈I Mi , and it will suffice to show that f is surjective, because this will imply that the larger set I ∪ {j} has the same property as I, thereby contradicting the maximality of I. T To show that f is surjective let g : U → U/ i∈I Mi ⊕ U/Mj and observe that T ( i∈I Mi ) + Mj = U since the left-hand side is strictly larger than Mj , which is maxT imal in U . Thus if x ∈ U we can write x = y + z where y ∈ i∈I Mi and z ∈ Mj . Now T T g(y) = (0, x + Mj ) and g(z) = (x + i∈I Mi , 0) so that both summands U/ i∈I Mi and U/Mj are contained in the image of g and g is surjective. Since f is obtained by composing T L g with the isomorphism which identifies U/ i∈I Mi with i∈I U/Mi , we deduce that f is surjective. (2) By the assumption that U has the descending chain condition on submodules, Rad U must be the intersection of finitely many maximal submodules. Therefore U/ Rad U is semisimple by part (1). If V is a submodule such that V is semisimple, say U/V ∼ = projection S1 ⊕· · ·⊕Sn where the Si are simple modules, let Mi be the kernel of U → U/V −→ Si . Then Mi is maximal and V = M1 ∩ · · · ∩ Mn . Thus V ⊇ Rad U , and Rad U is contained in every submodule V for which U/V is semisimple. As a particular case, we define the radical of a ring A to be the radical of the regular representation Rad A A and write simply Rad A. Thus by 6.6, if A has the descending chain condition on left ideals (for example, if A is a finite-dimensional algebra over a field) then Rad A is the smallest left ideal of A such that A/ Rad A is a semisimple module. (6.7) PROPOSITION. Let A be a ring. Then, (1) Rad A = {a ∈ A a · S = 0 for every simple A-module S}, and (2) Rad A is a 2-sided ideal of A. (3) Suppose further that A is a finite-dimensional algebra over a field. Then (a) Rad A is the smallest left ideal of A such that A/ Rad A is a semisimple A-module, (b) A is semisimple if and only if Rad A = 0, (c) Rad A is nilpotent, and is the largest nilpotent ideal of A. Proof. (1) Given a simple module S and 0 6= s ∈ S, the module homomorphism A → S given by a 7→ as is surjective and its kernel is a maximal left ideal Ms . Now if A a ∈ Rad A then a ∈ Ms for every S and s ∈ S, so as = 0 and a annihilates every simple module. Conversely, if a · S = 0 for every simple module S and M is a maximal left ideal then A/M is a simple module. Therefore a · (A/M ) = 0, which means a ∈ M . Hence T a ∈ maximalM M = Rad A. (2) Being the intersection of left ideals, Rad A is also a left ideal of A. Suppose that a ∈ Rad A and b ∈ A, so a · S = 0 for every simple S. Now a · bS ⊆ a · S = 0 so ab has the same property that a does. (3) (a) and (b) are immediate from 6.6. We prove (c). Choose any composition series 0 = An ⊂ An−1 ⊂ · · · ⊂ A1 ⊂ A A

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of the regular representation. Since each Ai /Ai+1 is a simple A-module, Rad A · Ai ⊆ Ai+1 by part (1). Hence (Rad A)r · A ⊆ Ar and (Rad A)n = 0. Suppose now that I is a nilpotent ideal of A, say I m = 0, and let S be any simple A-module. Then 0 = I m · S ⊆ I m−1 · S ⊆ · · · ⊆ IS ⊆ S is a chain of A-submodules of S, which are either 0 or S since S is simple. There must be some point where 0 = I r S 6= I r−1 S = S. Then IS = I · I r−1 S = I r S = 0, so in fact that point was the very first step. This shows that I ⊆ Rad A by part (1). Hence Rad A contains every nilpotent ideal of A, so is the unique largest such ideal. We may now determine the radical of kG when k is a field of characteristic p and G is a p-group. If R is any commutative ring, the ring homomorphism ǫ : RG → R g 7→ 1 for all g ∈ G is called the augmentation map. Regarded as a homomorphism of modules it expresses the trivial representation as a homomorphic image of the regular representation. The kernel of ǫ is called the augmentation ideal, and is denoted IG. Evidently IG consists of those P P elements g∈G ag g ∈ RG such that g∈G ag = 0. (6.8) PROPOSITION. (1) Let R denote the trivial RG-module. Then IG = {x ∈ RG x · R = 0}. (2) IG is free as an R-module with basis {g − 1 1 6= g ∈ G}. (3) If R = k is a field of characteristic p and G is a p-group then IG = Rad(kG). It follows that IG is nilpotent in this case. Proof. (1) The augmentation map ǫ is none other than the linear extension to RG of the homomorphism ρ : G → GL(1, R) which is the trivial representation. Thus each x ∈ RG acts on R as multiplication by ǫ(x), and so will act as 0 precisely if ǫ(x) = 0. (2) The elements g − 1 where g ranges through the non-identity elements of G are linearly independent since the elements g are, and they lie in IG. We show that they span P P IG. Suppose g∈G ag g ∈ IG, which means that g∈G ag = 0 ∈ R. Then X

g∈G

ag g =

X

g∈G

ag g −

X

g∈G

ag 1 =

X

ag (g − 1)

16=g∈G

is an expression as a linear combination of elements g − 1. (3) When G is a p-group and char(k) = p we have seen in 6.3 that k is the only simple kG-module. The result follows by part (1) and 6.7.

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Working in the generality of a finite-dimensional algebra A again, the radical of A allows us to give a further description of the radical and socle of a module. (6.9) PROPOSITION. Let A be a finite-dimensional algebra over a field k, and U an A-module. (1) The following are all descriptions of Rad U : (a) the intersection of the maximal submodules of U , (b) the smallest submodule of U with semisimple quotient, (c) Rad A · U . (2) The following are all descriptions of Soc U : (a) the sum of the simple submodules of U , (b) the largest semisimple submodule of U , (c) {u ∈ U Rad A · u = 0}.

Proof. We have seen the equivalence of descriptions (a) and (b) in 1.10 and 6.6. Let us show that the submodule Rad A · U in (1)(c) satisfies condition (1)(b). Firstly U/(Rad A·U ) is a module for A/ Rad A, which is a semisimple algebra. Hence U/(Rad A·U ) is a semisimple module and so Rad A · U contains the submodule of (1)(b). On the other hand if V ⊆ U is a submodule for which U/V is semisimple then Rad A · (U/V ) = 0 by 6.7, so V ⊇ Rad A · U . In particular, the submodule of (1)(b) contains Rad A · U . As for {u ∈ U Rad A · u = 0}, this is the largest submodule of U annihilated by Rad A. It is thus an A/ Rad A-module and hence is semisimple. Since every semisimple submodule of U is annihilated by Rad A, it is the unique largest such submodule. (6.10) Example. Consider the situation of 6.1 and 6.2 in which G is a cyclic group of order pn and k is a field of characteristic p. We see that Rad Ur ∼ = Ur−1 and Soc Ur ∼ = U1 n for 1 ≤ r ≤ p , taking U0 = 0. We now iterate the notions of socle and radical and for each A-module U we define inductively Radn (U ) = Rad(Radn−1 (U )) Socn (U )/ Socn−1 (U ) = Soc(U/ Socn−1 U ). It is immediate from 6.9 that Radn (U ) = (Rad A)n · U Socn (U ) = {u ∈ U (Rad A)n · u = 0}

and these submodules of U form chains

· · · ⊆ Rad2 U ⊆ Rad U ⊆ U 0 ⊆ Soc U ⊆ Soc2 U ⊆ · · · which are called, respectively, the radical series and socle series of U . They are also known as the lower Loewy series and upper Loewy series of U .

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(6.11) PROPOSITION. Let A be a finite-dimensional algebra over a field k, and U an A-module. The radical series of U is the fastest descending series of submodules of U with semisimple quotients, and the socle series of U is the fastest ascending series of U with semisimple quotients. The two series terminate, and if m and n are the least integers for which Radm U = 0 and Socn U = U then m = n. Proof. Suppose that · · · ⊆ U2 ⊆ U1 ⊆ U0 = U is a series of submodules of U with semisimple quotients. We show by induction on r that Radr (U ) ⊆ Ur . This is true when r = 0. Suppose that r > 0 and Radr−1 (U ) ⊆ Ur−1 . Then Radr−1 (U )/(Radr−1 (U ) ∩ Ur ) ∼ = (Radr−1 +Ur )/Ur ⊆ Ur−1 /Ur is semsimple, so Radr−1 (U )∩Ur ⊇ Rad(Radr−1 (U )) = Radr (U ). Therefore Radr (U ) ⊆ Ur . This shows that the radical series descends faster than the series Ui . The argument that the socle series ascends faster is similar. Since A is a finite-dimensional algebra we have (Rad A)r = 0 for some r. Then Radr U = 0 and Socr U = U by 6.9. By what we have just proved, the radical series descends at least as fast as the socle series and so has equal or shorter length. By a similar argument (using the fact that the socle series is the fastest ascending series with semisimple quotients) the socle series ascends at least as fast as the radical series and so has equal or shorter length. We conclude that the two lengths are equal. The common length of the radical series and socle series of U is called the Loewy length of the module U . We conclude this section by mentioning without proof the theorem of Jennings on the Loewy series of kG when G is a p-group and k is a field of characteristic p and summarize its implications. For proofs, see Benson’s book [Ben]. Jennings constructs a decreasing series of subgroups κr (G) = {g ∈ G g ≡ 1 modulo Radr (kG)}, which is sometimes called the Jennings series of G. This series of subgroups has the properties 1) [κr , κs ] ⊆ κr+s , 2) g p ∈ κip for all g ∈ κi , 3) κr /κ2r is elementary abelian. Furthermore, we may generate κr as (p)

κr = h[κr−1 , G], κ⌈r/p⌉ i, (p)

where κ1 = G, ⌈r/p⌉ is the least integer greater than or equal to r/p, and κr is the set of pth powers of elements of κr . Evidently the first term in this series is κ1 (G) = G and we may see that the second term κ2 (G) is the Frattini subgroup of G (the smallest normal

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subgroup of G for which the quotient is elementary abelian). After that the terms need to be calculated on a case-by-case basis. For each i ≥ 1 let di be the dimension of κi /κi+1 as a vector space over Fp , and choose any elements xi,s of G such that the set {xi,s κi+1 1 ≤ s ≤ di } forms a basis for κi /κi+1 . Q αi,s Let x ¯i,s = xi,s − 1 ∈ kG. There are |G| products of the form x ¯i,s , where the factors are taken in some predetermined order, and 0 ≤ αi,s ≤ p − 1. The weight of such a product is P defined to be iαi,s . Jenning’s theorem states that the set of products of weight w lie in w Rad (kG), and forms a basis of Radw (kG) modulo Radw+1 (kG). Thus the set of all these products is a basis for kG compatible with the powers of the radical. For example, if we take an element x of order 4 and an element y of order 2 which generate the dihedral group of order 8, so D8 = hx, y x4 = y 2 = 1, yxy = x−1 i, we have κ1 = D8 , κ2 = hx2 i, κ3 = 1. We may choose x1,1 = x, x1,2 = y, x2,1 = x2 , and note that x ¯2 = x¯2 . Now the products x ¯α1,1 x ¯2α2,1 y¯α1,2 = x ¯α1,1 +2α2,1 y¯α1,2 , where 0 ≤ αi,s ≤ 1, form a basis of kD8 which is compatible with the powers of the radical. In this special case, these elements may be simplified as x ¯i y¯j with 0 ≤ i ≤ 3 and 0 ≤ j ≤ 1. When doing calculation with group rings of p-groups in characteristic p the basis given by Jennings is often to be preferred over the basis given by the group elements. This basis gives a description of the powers of the radical in group-theoretic terms, and it allows us to deduce a result about the socle series as well. Since each element x ¯i,s of weight i contributes factors of weights 0, i, 2i, 3i, . . . , (p − 1)i in Jennings’ basis , the total number of products of weight w is the coefficient of tw in 2

(1 + t + t + · · · + t

p−1 d1

2

) (1 + t + · · · + t

2(p−1) d2

)

··· =

Y  (1 − tip ) di

i≥1

(1 − ti )

,

P and this equals the polynomial w≥0 (dim Radw (kG)/ dim Radw+1 (kG))tw . We see from this that the dimensions of the factors in the radical series of kG are symmetric, in that they are the same if taken in reverse order. Jumping ahead of ourselves for a moment and using the fact that kG ∼ = kG∗ (see Section 8) and also using Exercise 6 to this section, we may see that when G is a p-group and k is a field of characteristic p the terms of the radical series and the socle series of kG coincide, although these terms appear in the two series in the opposite order. Exercises for Section 6. 1. Let A be a ring. Prove that for each n, Socn A A is a 2-sided ideal of A. P 2. Let G = g∈G g as an element of kG, where k is a field. Show that the subspace kG of kG spanned by G is an ideal. Show that this ideal is nilpotent if and only if the characteristic of k divides |G|. Deduce that if kG is semisimple then char(k)6 |G|. 3. Prove that if N is a normal subgroup of G and k is a field then Rad(kN ) = kN ∩ Rad(kG).

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4. Suppose that U is an indecomposable module with just two composition factors. Show that U is uniserial. 5. Show that the following conditions are equivalent for a module U which has a composition series. (a) U is uniserial (i.e. U has a unique composition series). (b) The set of all submodules of U is totally ordered by inclusion. (c) Radr U/ Radr+1 U is simple for all r. (d) Socr+1 U/ Socr U is simple for all r. 6. Let U be a kG-module and U ∗ its dual. Show that for each n Socn (U ∗ ) = {f ∈ U ∗ f (Radn (U ) = 0} and

Radn (U ∗ ) = {f ∈ U ∗ f (Socn (U ) = 0}.

Deduce that Socn+1 (U ∗ )/ Socr (U ∗ ) ∼ = (Radn (U )/ RadN+1 (U ))∗ as kG-modules. [Hint: recall Exercise 9 to Section 4.] 7. Let H be a subgroup of G. P (a) Let H = h∈H h be the sum of the elements of H, as an element of RG. Show that RG · H ∼ = R ↑G H as RG-modules. (b) Let IH be the augmentation ideal of RH, as a subset of RG. Show that RG·IH ∼ = G G ∼ IH ↑H as RG-modules, and that RG/(RG · IH) = R ↑H as RG-modules. The next five exercises give a direct proof of the result stated in 6.8, that for a p-group in characteristic p the augmentation ideal is nilpotent. 8. Show that if elements g1 , . . . , gn generate G as a group, then (g1 − 1), . . . , (gn − 1) generate the augmentation ideal IG as a left ideal of kG. [Use the formula (gh − 1) = g(h − 1) + (g − 1).] 9. Suppose that k is a field of characteristic p and G is a p-group. Prove that each element (g − 1) is nilpotent. (More generally, every element of IG is nilpotent.) 10. Show that if N is a normal subgroup of G then the left ideal RG · IN = {x · y x ∈ RG, y ∈ IN }

of RG generated by IN is the kernel of the ring homomorphism RG → R[G/N ] and is in fact a 2-sided ideal in RG. [One approach to this uses the formula g(n − 1) = ( gn − 1)g.] Show that (RG · IN )r = RG · (IN )r for all r. 11. Show that if a particular element (g − 1) appears n times in a product (g1 − 1) · · · (gr − 1) then (g1 − 1) · · · (gr − 1) ≡ (g − 1)n · x

modulo kG · (IG′ )

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for some x ∈ kG, where G′ denotes the commutator subgroup. [Use the formula (g − 1)(h − 1) = (h − 1)(g − 1) + (ghg −1 h−1 − 1)hg.] Show that if G is a p-group and k a field of characteristic p then IGr ⊆ kG · IG′ for some power r. 12. Prove that if G is a p-group and k is a field of characteristic p then (IG)r = 0 for some power r. 13. Calculate the Loewy length of kCpn , the group algebra of the direct product of n copies of a cycle of order p. 14. The dihedral group of order 2n has a presentation D2n = hx, y x2 = y 2 = (xy)n = 1i.

Let k be a field of characteristic 2. Show that when n is a power of 2, each power (ID2n )r of the augmentation ideal is spanned modulo (ID2n )r+1 by the two products (x − 1)(y − 1)(x − 1)(y − 1) · · · and (y − 1)(x − 1)(y − 1)(x − 1) · · · of length r. Hence calculate the Loewy length of kD2n and show that Rad(kD2n )/ Soc(D2n ) is the direct sum of two kD2n -modules which are uniserial. 15. When n ≥ 3, the generalized quaternion group of order 2n has a presentation n−1 n−2 = 1, y 2 = x2 , yxy −1 = x−1 i. Q2n = hx, y x2

Let k be a field of characteristic 2. Show that when r ≥ 1 each power (IQ2n )r of the augmentation ideal is spanned modulo (IQ2n )r+1 by (x − 1)r and (x − 1)r−1 (y − 1). Hence calculate the Loewy length of kQ2n . 16. Show that for each RG-module U , U/(IG · U ) is the largest quotient of U on which G acts trivially. Prove that U/(IG · U ) ∼ = R ⊗RG U . 17. (a) Let G be any group and IG ⊂ ZG the augmentation ideal over Z. Prove that IG/(IG)2 ∼ = G/G′ as abelian groups. [Consider the homomorphism of abelian groups IG → G/G′ given by g − 1 7→ gG′ . Use the formula ab − 1 = (a − 1) + (b − 1) + (a − 1)(b − 1) to show that (IG)2 is contained in the kernel, and that the homomorphism G/G′ → IG/(IG)2 given by gG′ 7→ g − 1 + (IG)2 is well defined.] (b) If now R is any commutative ring with 1 and IG ⊂ RG is the augmentation ideal, show that IG/(IG)2 ∼ = R ⊗Z G/G′ as R-modules. 18. Let Ω be a transitive G-set and k a field. Let kΩ be the corresponding permutation P module. There is a homomorphism of kG-modules ǫ : kΩ → k defined as ǫ( ω∈Ω aω ω) = P P ω∈Ω aω . Let Ω = ω∈Ω ω ∈ kΩ. (a) Show that every kG-module homomorphism kΩ → k is a scalar multiple of ǫ. G (b) Show that the fixed points of G on kΩ are kΩ = k · Ω. (c) Show that ǫ(Ω) = 0 if and only if char k |Ω|, and that if this happens then Ω ∈ Rad kΩ and the trivial module k occurs as a composition factor of kΩ with multiplicity ≥ 2.

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(d) Show that if ǫ(Ω) 6= 0 then ǫ is a split epimorphism and Ω 6∈ Rad kΩ. (e) Show that kG is semisimple if and only if the regular representation kG has the trivial module k as a direct summand (i.e. k is a projective module). 19. Let Ω be a transitive G-set for a possibly infinite group G and let RΩ be the corresponding permutation module. Show that Ω is infinite if and only if (RΩ)G = 0 and deduce that G is infinite if and only if (RG)G = 0. 20. Let Ur be the indecomposable kCp -module of dimension r, 1 ≤ r ≤ p, where k is a field of characteristic p. Prove that Ur = S r−1 (U2 ), the (r − 1) symmetric power. [One way to proceed is to show that if Cp = hgi then (g − 1)r−1 does not act as zero on S r−1 (U2 ) and use the classification of indecomposable kCp -modules.] 21. Let G = SL(2, p), the group of 2 × 2 matrices over Fp which have determinant 1, where p is a prime. The subgroups 

1 P1 = { 0

λ 1



λ ∈ Fp },



1 P2 = { λ

0 1



λ ∈ Fp }

have order p. Let U2 be the natural 2-dimensional module on which G acts. When 0 ≤ r ≤ p − 1 prove that S r (U2 ) is a uniserial Fp P1 -module, and also a uniserial Fp P2 module, but that the only subspaces of S r (U2 ) which are invariant under both P1 and P2 are 0 and S r (U2 ). Deduce that S r (U2 ) is a simple Fp G-module. [Background to the question which is not needed to solve it: |G| = p(p2 − 1); both P1 and P2 are Sylow p-subgroups of G. In fact, the simple modules constructed here form a complete list of the simple Fp SL(2, p)-modules.] 22. Let g be an endomorphism of a finite-dimensional vector space V over a field k of characteristic p, and suppose that g has finite order pd for some d. Show that as a khgi-module, V has an indecomposable direct summand of dimension at least pd−1 + 1. [You may assume the classification of indecomposable modules for cyclic p-groups in characteristic p.] Deduce that if such an endomorphism g fixes a subspace of V of codimension 1 then g has order p or 1. [An endomorphism (not necessarily of prime-power order) which fixes a subspace of codimension 1 is sometimes referred to as a reflection in a generalized sense.]

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7. Projective modules for finite-dimensional algebras In previous sections we have seen the start of techniques to describe modules which are not semisimple. The most basic decomposition of such a module is one that expresses it as a direct sum of modules which cannot be decomposed as a direct sum any further. These summands are called indecomposable modules. We have also examined the notions of radical series and socle series of a module, which are series of canonically defined submodules which may shed light on submodule structure. We combine these two notions in the study of projective modules for group rings, working at first in the generality of modules for finite dimensional algebras over a field. In this situation the indecomposable projective modules are the indecomposable summands of the regular representation. We will see that they are identified by the structure of their radical quotient. The projective modules are important because their structure is part of the structure of the regular representation. Since every module is a homomorphic image of a direct sum of copies of the regular representation, by knowing the structure of the projectives one knows, in some sense, all modules. Recall that a module M over a ring A is said to be free if it has a basis, that is, a subset which spans M as an A-module, and is linearly independent over A. Whenever L ∼ {xi i ∈ I} is a basis it follows that M = i∈I Axi with Axi = A for all i. Thus M is a finitely generated free module if and only if M ∼ = An for some n, that is, M may be identified with the module of n-tuples of elements of A. (7.1) PROPOSITION. Let A be a ring and M an A-module. Then M is free on a subset {xi i ∈ I} if and only if for every module N and mapping of sets φ : {xi i ∈ I} → N there exists a unique morphism of modules ψ : M → N which extends φ. We omit the proof. We define a module homomorphism f : M → N to be a split epimorphism if and only if there exists a homomorphism g : N → M so that f g = 1N , the identity map on N . Note that a split epimorphism is necessarily an epimorphism since if x ∈ N then x = f (g(x)) so that x lies in the image of f . We define similarly f to be a split monomorphism if there exists a homomorphism g : N → M so that gf = 1M . Necessarily a split monomorphism is a monomorphism. We are about to show that if f is a split epimorphism then N is (isomorphic to) a direct summand of M . To combine both this and the corresponding result for split monomorphisms it is convenient to introduce short exact sequences. We β α say that a diagram of modules and module homomorphisms L−→M −→N is exact at M β α if Im α = Ker β. A short exact sequence of modules is a diagram 0 → L−→M −→N → 0 which is exact a each of L, M and N . Exactness at L and N means simply that α is a monomorphism and β is an epimorphism.

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β

α

(7.2) PROPOSITION. Let 0 → L−→M −→N → 0 be a short exact sequence of modules over a ring. The following are equivalent: (1) α is a split monomorphism, (2) β is a split epimorphism, and (3) there is a commutative diagram 0 → L

α

−→

k 0 → L

ι

M  γ y

1 −→ L⊕N

β

−→ N

→ 0

k π

2 −→ N

→ 0

where ι1 and π2 are inclusion into the first summand and projection onto the second summand. In any diagram such as the one in (3) the morphism γ is necessarily an isomorphism. Thus if any of the three conditions is satisfied it follows that M ∼ = L ⊕ N. Proof. Condition (3) implies the first two, since the existence of such a commutative diagram implies that α is split by π1 γ and β is split by γ −1 ι2 . Conversely if condition (1) is satisfied, so that δα = 1L for some homomorphism δ : M → L, we obtain a commutative diagram as in (3) on taking the components of γ to be δ and β. If condition (2) is satisfied we obtain a commutative diagram similar to the one in (3) but with a homomorphism ζ : L ⊕ N → M in the wrong direction, whose components are α and a splitting of β. We obtain the diagram of (3) on showing that in any such diagram the middle vertical homomorphism must be invertible. The fact that the middle homomorphism in the diagram must be invertible is a consequence of both the ‘five lemma’ and the ‘snake lemma’ in homological algebra. We leave it here as an exercise. In the event that α and β are split, we say that the short exact sequence in Proposition 7.2 is split. Notice that whenever β : M → N is an epimorphism it is part of the short exact β sequence 0 → Ker β ֒→ M −→N → 0 and so we deduce that if β is a split epimorphism then N is a direct summand of M . A similar comment evidently applies to split monomorphisms. (7.3) PROPOSITION. The following are equivalent for an A-module U . (1) U is a direct summand of a free module. (2) Every epimorphism V → U is split. (3) For every pair of morphisms U  α y V

β

−→

W

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where β is an epimorphism, there exists a morphism γ : U → V with βγ = α. (4) For every short exact sequence of A-modules 0 → V → W → X → 0 the corresponding sequence 0 → HomA (U, V ) → HomA (U, W ) → HomA (U, X) → 0 is exact. Proof. This result is standard and we do not prove it here. In condition (4) the sequence of homomorphism groups is always exact at the left-hand terms HomA (U, V ) and HomA (U, W ) without requiring any special property of U (we say that HomA (U, ) is left exact). The force of condition (4) is that the sequence should be exact at the right-hand term. We say that a module U satisfying any of the four conditions of 7.3 is projective. Notice that direct sums and also direct summands of projective modules are projective. An indecomposable module which is projective is an indecomposable projective module, and these modules will be very important in our study. In other texts the indecomposable projective modules are also known as PIMs, or Principal Indecomposable Modules, but we will not use this terminology here. One way to obtain projective A-modules is from idempotents of the ring A. If e2 = e ∈ A then A A = Ae ⊕ A(1 − e) as A-modules, and so the submodules Ae and A(1 − e) are projective. We formalize this with the next result, which should be compared with 3.22 in which we were dealing with ring summands of A and central idempotents. (7.4) PROPOSITION. Let A be a ring. The decompositions of the regular representation as a direct sum of submodules AA

= A1 ⊕ · · · ⊕ Ar

biject with expressions 1 = e1 + · · · + er for the identity of A as a sum of orthogonal idempotents, in such a way that Ai = Aei . The summand Ai is indecomposable if and only if the idempotent ei is primitive. Proof. Suppose that 1 = e1 + · · · + er is an expression of the identity as a sum of orthogonal idempotents. Then AA

= Ae1 ⊕ · · · ⊕ Aer ,

for the Aei are evidently submodules of A, and their sum is A since if x ∈ A then x = P xe1 + · · · + xer . The sum is direct since if x ∈ Aei ∩ j6=i Aej then x = xei and also P P x = j6=i aj ej so x = xei = j6=i aj ej ei = 0. Conversely, suppose that A A = A1 ⊕ · · · ⊕ Ar is a direct sum of submodules. We may write 1 = e1 + · · · + er where ei ∈ Ai is a uniquely determined element. Now

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ei = ei 1 = ei e1 + · · · + ei er is an expression in which ei ej ∈ Aj , and since the only such expression is ei itself we deduce that ei ej =

n

ei 0

if i = j, otherwise.

The two constructions just described, in which we associate an expression for 1 as a sum of idempotents to a module direct sum decomposition and vice-versa, are mutually inverse, giving a bijection as claimed. If a summand Ai decomposes as the direct sum of two other summands, this gives rise to an expression for ei as a sum of two orthogonal idempotents, and conversely. Thus Ai is indecomposable if and only if ei is primitive. In 3.22 there was a statement about the uniqueness of summands in a decomposition of A as a direct sum of indecomposable rings, and we should point out that the corresponding uniqueness statement need not hold with module decompositions of A A. For an example of this we might take A = M2 (R), the ring of 2 × 2-matrices over a ring R, in which AA

=A



1 0 0 0



⊕A



0 0 0 1



=A



0 1 0 1



⊕A



1 0

−1 0



.

The submodules here are all different. We will see later that if A is a finite-dimensional algebra over a field then in any two decompositions of A A as a direct sum of indecomposable submodules, the submodules are isomorphic in pairs. We will also see that when A is a finite-dimensional algebra over a field, every indecomposable projective A-module may be realized as Ae for some primitive idempotent e. For other rings this need not be true, and an example is ZG, for which it is the case that the only idempotents are 0 and 1 (see the exercises to Section 8). For certain finite groups (an example is the cyclic group of order 23, but this takes us beyond the scope of this book) there exist indecomposable projective ZG-modules which are not free, so such modules will never have the form ZGe. (7.5) Example. We present an example of a decomposition of the regular representation in a situation which is not semisimple. Many of the observations we will make are consequences of theory to be presented in later sections, but it seems worthwhile to show that the calculations can be done by direct arguments. Consider the group ring F4 S3 where F4 is the field of 4 elements. By Proposition 4.5 the 1-dimensional representations of S3 are the simple representations of S3 /S3′ ∼ = C2 , lifted to S3 . But F4 C2 has only one simple module, namely the trivial module, by Proposition 6.3, so this is the only 1-dimensional F4 S3 -module. The 2-dimensional representation of S3 constructed in Section 1 over any coefficient ring is now seen to be simple here, since otherwise it would have a trivial submodule; but the eigenvalues of the element (1, 2, 3) on this module are ω and ω 2 , where ω ∈ F4 is a primitive cube root of 1, so there is no trivial submodule.

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Let K = h(1, 2, 3)i be the subgroup of S3 of order 3. Now F4 K is semisimple with three 1-dimensional representations on which (1, 2, 3) acts as 1, ω and ω 2 , respectively. In fact F4 K = F4 Ke1 ⊕ F4 Ke2 ⊕ F4 Ke3 where

e1 = () + (1, 2, 3) + (1, 3, 2) e2 = () + ω(1, 2, 3) + ω 2 (1, 3, 2) e3 = () + ω 2 (1, 2, 3) + ω(1, 3, 2)

are orthogonal idempotents in F4 K. We may see that these are orthogonal idempotents by direct calculation, but it can also be seen by observing that the corresponding elements 2πi of CK with ω replaced by e 3 are orthogonal and square to 3 times themselves (Theorem 2πi 2πi 3.23), and lie in Z[e 3 ]K. Reduction modulo 2 gives a ring homomorphism Z[e 3 ] → F4 which maps these elements to e1 , e2 and e3 , while retaining their properties. Thus F4 S3 = F4 S3 e1 ⊕ F4 S3 e2 ⊕ F4 S3 e3 and we have constructed modules F4 S3 ei which are projective. We have not yet shown that they are indecomposable. We easily compute that (1, 2, 3)e1 = e1 ,

(1, 2, 3)e2 = ω 2 e2 ,

(1, 2, 3)e3 = ωe3

and from this we see that K · F4 ei = F4 ei for all i. Since S3 = K ∪ (1, 2)K we have F4 S3 ei = F4 ei ⊕ F4 (1, 2)ei , which has dimension 2 for all i. We have already seen that when i = 2 or 3, ei is an eigenvector for (1, 2, 3) with eigenvalue ω or ω 2 , and a similar calculation shows that the same is true for (1, 2)ei . Thus when i = 2 or 3, F4 S3 ei has no trivial submodule and hence is simple by the observations made at the start of this example. We have an isomorphism of F4 S3 -modules F4 S3 e2 → F4 S3 e3 e2 7→ (1, 2)e3 (1, 2)e2 7→ e3 . P On the other hand F4 S3 e1 has fixed points F4 g∈S3 g of dimension 1 and so has two composition factors, which are trivial. On restriction to F4 h(1, 2)i it is the regular representation, and it is a uniserial module. We see from all this that F4 S3 = 11 ⊕ 2 ⊕ 2, in a diagrammatic notation. Thus the 2-dimensional simple F4 S3 -module is projective, and the trivial module appears as the unique simple quotient of a projective module of dimension 2 whose socle is also the trivial module. These summands of F4 S3 are indecomposable, and so e1 , e2 and e3 are primitive P idempotents in F4 S3 . We see also that the radical of F4 S3 is the span of g∈S3 g.

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We now develop the theory of projective covers. We first make the definition that an essential epimorphism is an epimorphism of modules f : U → V with the property that no proper submodule of U is mapped surjectively onto V by f . An equivalent formulation is that whenever g : W → U is a map such that f g is an epimorphism, then g is an epimorphism. One immediately asks for examples of essential epimorphisms, but it is probably more instructive to consider epimorphisms which are not essential. If U → V is any epimorphism and X is a non-zero module then the epimorphism U ⊕ X → V which is zero on X can never be essential because U is a submodule of U ⊕ X mapped surjectively onto V . Thus if U → V is essential then U can have no direct summands which are mapped to zero. One may think of an essential epimorphism as being minimal, in that no unnecessary parts of U are present. The greatest source of essential epimorphisms is Nakayama’s lemma. (7.6) THEOREM (Nakayama’s Lemma). If U is any Noetherian module, the homomorphism U → U/ Rad U is essential. Proof. Suppose V is a submodule of U . If V 6= U then V ⊆ M ⊂ U where M is a maximal submodule of U . Now V + Rad U ⊆ M and so the composite V → U → U/ Rad U has image contained in M/ Rad U , which is not equal to U/ Rad U since (U/ Rad U )/(M/ Rad U ) ∼ = U/M 6= 0. Evidently an equivalent way to state Nakayama’s lemma is that if V is a submodule of U with the property that V + Rad U = U , then V = U . As an example of an essential epimorphism we could consider the indecomposable kGmodule Ur of dimension r, where k is a field of characteristic p and G is cyclic of order pn . Since Ur has a unique maximal submodule, of codimension 1, the epimorphism Ur → U1 is essential, by Nakayama’s lemma. The next result is not at all difficult and could usefully be proved as an exercise. (7.7) PROPOSITION. (a) Suppose that f : U → V and g : V → W are two module homomorphsms. If two of f , g and gf are essential epimorphisms then so is the third. (b) Let f : U → V be a homomorphism of modules for a finite-dimensional algebra over a field. Then f is an essential epimorphism if and only if the homomorphism of radical quotients U/ Rad U → V / Rad V is an isomorphism. (c) Let fi : Ui → Vi be homomorphisms of modules for a finite-dimensional algebra over a field, where i = 1, . . . , n. The fi are all essential epimorphisms if and only if ⊕fi :

M i

Ui →

M i

Vi

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is an essential epimorphism. Proof. (a) Suppose f and g are essential epimorphisms. Then gf is an epimorphism also, and it is essential because if U0 is a proper submodule of U then f (U0 ) is a proper submodule of V since f is essential, and hence g(f (U0)) is a proper submodule of S since g is essential. Next suppose f and gf are essential epimorphisms. Since W = Im(gf ) ⊆ Im(g) it follows that g is an epimorphism. If V0 is a proper submodule of V then f −1 (V0 ) is a proper submodule of U since f is an epimorphism, and now g(V0 ) = gf (f −1 (V0 )) is a proper submodule of S since gf is essential. Suppose that g and gf are essential epimorphisms. If f were not an epimorphism then f (U ) would be a proper submodule of V , so gf (U ) would be a proper submodule of W since gf is essential. Since gf (U ) = W we conclude that f is an epimorphism. If U0 is a proper submodule of U then gf (U0 ) is a proper submodule of W , since gf is essential, so f (U0 ) is a proper submodule of V since g is an epimorphism. Hence f is essential. (b) Consider the commutative square U   y

U/ Rad U

−→

V   y

−→ V / Rad V

where the vertical homomorphisms are essential epimorphisms by Nakayama’s lemma. Now if either of the horizontal arrows is an essential epimorphism then so is the other, using part (a). The bottom arrow is an essential epimorphism if and only if it is an isomorphism; for U/ Rad U is a semisimple module and so the kernel of the map to V / Rad V has a direct complement in U/ Rad U , which maps onto V / Rad V . Thus if U/ Rad U → V / Rad V is an essential epimorphism its kernel must be zero and hence it must be an isomorphism. (c) The map (⊕i Ui )/ Rad(⊕i Ui ) → (⊕i Vi )/ Rad(⊕i Vi ) induced by ⊕fi may be identified as a map M M (Ui / Rad Ui ) → (Vi / Rad Vi ), i

i

and it is an isomorphism if and only if each map Ui / Rad Ui → Vi / Rad Vi is an isomorphism. These conditions hold if and only if ⊕fi is an essential epimorphism, if and only if each fi is an essential epimorphism by part (b).

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We define a projective cover of a module U to be an essential epimorphism P → U , where P is a projective module. Strictly speaking the projective cover is the homomorphism, but we may also refer to the module P as the projective cover of U . We are justified in calling it the projective cover by the second part of the following result, which says that projective covers (if they exist) are unique. (7.8) PROPOSITION. (1) Suppose that f : P → U is a projective cover of a module U and g : Q → U is an epimorphism where Q is a projective module. Then we may write Q = Q1 ⊕ Q2 so that g has components g = (g1 , 0) with respect to this direct sum decomposition and g1 : Q1 → U appears in a commutative triangle Q1   g1 y

γ

ւ P

f

−→

U

where γ is an isomorphism. (2) If any exist, the projective covers of a module U are all isomorphic, by isomorphisms which commute with the essential epimorphisms. Proof. (1) In the diagram

P

f

Q  g y

−→ U

we may lift in both directions to obtain maps α : P → Q and β : Q → P so that the two triangles commute. Now f βα = gα = f is an epimorphism, so βα is also an epimorphism since f is essential. Thus β is an epimorphism. Since P is projective β splits and Q = Q1 ⊕ Q2 where Q2 = Ker β, and β maps Q1 isomorphically to P . Thus g = (f β|Q1 , 0) is as claimed with γ = β|Q1 . (2) Supposing that f : P → U and g : Q → U are both projective covers, since Q1 is a submodule of Q which maps onto U and f is essential we deduce that Q = Q1 . Now γ : Q → P is the required isomorphism. (7.9) COROLLARY. If P and Q are Noetherian projective modules over a ring then P ∼ = Q if and only if P/ Rad P ∼ = Q/ Rad Q. Proof. By Nakayama’s lemma P and Q are the projective covers of P/ Rad P and Q/ Rad Q. It is clear that if P and Q are isomorphic then so are P/ Rad P and Q/ Rad Q, and conversely if these quotients are isomorphic then so are their projective covers, by uniqueness of projective covers.

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If now P and Q are projective modules for a finite-dimensional algebra A over a field the radical quotients P/ Rad P and Q/ Rad Q are semisimple, and to classify the indecomposable projectives it will suffice to classify these semisimple quotients, by Corollary 7.9. We will see in this case that if P is an indecomposable projective A-module then it is isomorphic to a module Af for some primitive idempotent f ∈ A, and the quotient P/ Rad P is isomorphic to (A/ Rad A)e for some primitive idempotent e of A/ Rad A where e = f + Rad A. In general if I is an ideal of a ring A and f is an idempotent of A then clearly f + I is also an idempotent of A/I. On the other hand, given an idempotent e of A/I, if we can always find an idempotent f ∈ A such that e = f + I we say we can lift idempotents from A/I to A. We state the next results about lifting idempotents in the context of a ring with a nilpotent ideal I, but readers familiar with completions will recognize that these results extend to a situation where A is complete with respect to the I-adic topology on A. (7.10) THEOREM. Let I be a nilpotent ideal of a ring A and e an idempotent in A/I. Then there exists an idempotent f ∈ A with e = f + I. If e is primitive, so is f . Proof. We define idempotents ei ∈ A/I i inductively such that ei + I i−1 /I i = ei−1 for all i, starting with e1 = e. Suppose that ei−1 is an idempotent of A/I i−1 . Pick any element a ∈ A/I i mapping onto ei−1 , so that a2 − a ∈ I i−1 /I i . Since (I i−1 )2 ⊆ I i we have (a2 − a)2 = 0 ∈ A/I i . Put ei = 3a2 − 2a3 . This does map to ei−1 ∈ A/I i−1 and we have e2i − ei = (3a2 − 2a3 )(3a2 − 2a3 − 1) = −(3 − 2a)(1 + 2a)(a2 − a)2 = 0. This completes the inductive definition, and if I r = 0 we put f = er . Suppose that e is primitive and that f can be written f = f1 + f2 where f1 and f2 are orthogonal idempotents. Then e = e1 + e2 , where ei = fi + I, is also a sum of orthogonal idempotents. Therefore one of these is zero, say, e1 = 0 ∈ A/I. This means that f12 = f1 ∈ I. But I is nilpotent, and so contains no non-zero idempotent. We will very soon see in the situation of 7.10 that it is also true that if f is primitive, so is e. (7.11) COROLLARY. Let I be a nilpotent ideal of a ring A and let 1 = e1 + · · · + en be a sum of orthogonal idempotents in A/I. Then we can write 1 = f1 + · · · + fn in A, where the fi are orthogonal idempotents such that fi + I = ei for all i. If the ei are primitive then so are the fi . Proof. We proceed by induction on n, the induction starting when n = 1. Suppose that n > 1 and the result holds for smaller values of n. We will write 1 = e1 + E in A/I where

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E = e2 + · · ·+ en is an idempotent orthogonal to e1 , and by Theorem 7.10 we may lift e1 to an idempotent f1 ∈ A. Write F = 1 − f1 , so that F is an idempotent which lifts E. Now F is the identity element of the ring F AF which has a nilpotent ideal F IF . The composite homomorphism F AF ֒→ A → A/I has kernel F AF ∩ I and this equals F IF , since clearly F AF ∩ I ⊇ F IF , and if x ∈ F AF ∩ I then x = F xF ∈ F IF , so F AF ∩ I ⊆ F IF . Inclusion of F AF in A thus induces a monomorphism F AF/F IF → A/I, and its image is E(A/I)E. In E(A/I)E the identity element E is the sum of n − 1 orthogonal idempotents, and this expression is the image of a similar expression for F + F IF in F AF/F IF . By induction, there is a sum of orthogonal idempotents F = f2 + · · · + fn in F AF which lifts the expression in F AF/F IF and hence also lifts the expression for E in A/I, so we have idempotents fi ∈ A, i = 1, . . . , n with fi + I = ei . These fi are orthogonal; for f2 , . . . , fn are orthogonal in F AF by induction, and if i > 1 then F fi = fi so we have f1 fi = f1 F fi = 0. The final assertion about primitivity is the last statement of 7.10. (7.12) COROLLARY. Let f be an idempotent in a ring A which has a nilpotent ideal I. Then f is primitive if and only if f + I is primitive. Proof. We have seen in 7.10 that if f + I is primitive, then so is f . Conversely, if f + I can be written f + I = e1 + e2 where the ei are orthogonal idempotents of A/I, then by applying 7.11 to the ring f Af (of which f is the identity) we may write f = g1 + g2 where the gi are orthogonal idempotents of A which lift the ei . We now classify the indecomposable projective modules over a finite-dimensional algebra as the projective covers of the simple modules. We first describe how these projective covers arise, and then show that they exhaust the possibilities for indecomposable projective modules. We postpone explicit examples until the next section, in which we consider group algebras. (7.13) THEOREM. Let A be a finite-dimensional algebra over a field. For each simple module S there is an indecomposable projective module PS with PS / Rad PS ∼ = S. It follows that PS is the projective cover of S, and is uniquely determined up to isomorphism by this property. The projective module PS has the form PS = Af where f is a primitive idempotent with the property that f S 6= 0, and if T is any simple module not isomorphic to S then f T = 0. Proof. Let e ∈ A/ Rad A be any primitive idempotent such that eS 6= 0, and let f be any lift of e to A. Then f S 6= 0 and f is primitive. We define PS = Af , an indecomposable projective module. Now PS / Rad PS = Af / Rad A · Af ∼ = (A/ Rad A) · (f + Rad A) = S

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the isomorphism arising because the map Af → (A/ Rad A) · (f + Rad A) defined by af 7→ (af + Rad A) has kernel (Rad A) · f . We have f S = eS 6= 0 and f T = eT = 0 if T ∼ 6 S since a primitive idempotent e in the semisimple ring A/ Rad A is non-zero on a = unique isomorphism class of simple modules. (7.14) THEOREM. Let A be a finite-dimensional algebra over a field k. Up to isomorphism, the indecomposable projective A-modules are exactly the modules PS which are the projective covers of the simple modules, and PS ∼ = PT if and only if S ∼ = T . Each projective PS appears as a direct summand of the regular representation, with multiplicity equal to the multiplicity of S as a summand of kG/ Rad(kG). If k is algebraically closed we have M A∼ (PS )dim S . = simple S

Proof. Let P be an indecomposable projective module and write P/ Rad P ∼ = S1 ⊕ · · · ⊕ Sn . Then P → S1 ⊕ · · · ⊕ Sn is a projective cover. Now PS1 ⊕ · · · ⊕ PSn → S1 ⊕ · · · ⊕ Sn is also a projective cover, and by uniqueness of projective covers we have P ∼ = PS1 ⊕ · · · ⊕ P Sn . Since P is indecomposable we have n = 1 and P ∼ = PS1 . Suppose that each simple A module S occurs with multiplicity nS as a summand L nS of the semisimple ring A/ Rad A. Both A and are the projective cover of simple S PS A/ Rad A, and so they are isomorphic. It is always the case that nS 6= 0, and when k is algebraically closed nS = dim S. (7.15) THEOREM. Let A be a finite-dimensional algebra over a field k, and U a finitely-generated A-module. Then U has a projective cover. Proof. Since U/ Rad U is semisimple we may write U/ Rad U = S1 ⊕ · · · ⊕ Sn , where the Si are simple modules. Let PSi be the projective cover of Si and h : PS1 ⊕ · · · ⊕ PSn → U/ Rad U the projective cover of U/ Rad U . By projectivity there exists a homomorphism f such that the following diagram commutes: f

ւ U

g

−→

PS1 ⊕ · · · ⊕ P Sn   . yh U/ Rad U

Since both g and h are essential epimorphisms, so is f by 7.7. Therefore f is a projective cover.

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We should really learn more from 7.15 than simply that U has a projective cover: the projective cover of U is the same as the projective cover of U/ Rad U . (7.16) Example. The arguments which show the existence of projective covers have a sense of inevitability about them and we may get the impression that projective covers always exist in arbitrary situations. In fact they fail to exist in general for integral group rings. If G = {e, g} is a cyclic group of order 2, consider the submodule 3Z · e + Z · (e + g) of ZG generated as an abelian group by 3e and e + g. We rapidly check that this subgroup is invariant under the action of G, and is a ZG-submodule, and it is not the whole of ZG since it does not contain e. Applying the augmentation map ǫ : ZG → Z we have ǫ(3e) = 3 and ǫ(e + g) = 2 so ǫ(3Z · e + Z · (e + g)) = 3Z + 2Z = Z, and thus ǫ is an epimorphism which is not essential. If Z were to have a projective cover it would be a proper summand of ZG by 7.8. On reducing modulo 2 we would deduce that F2 G decomposes, which we know not to be the case. Now that we have classified the projective modules for a finite-dimensional algebra we turn to one of their important uses, which is to determine the multiplicity of a simple module S as a composition factor of an arbitrary module U (with a composition series). If 0 = U0 ⊂ U1 ⊂ · · · ⊂ Un = U is any composition series of U , the number of quotients Ui /Ui−1 isomorphic to S is determined independently of the choice of composition series, by the Jordan-H¨older theorem. We call this number the (composition factor) multiplicity of S in U . (7.17) PROPOSITION. Let S be a simple module for with projective cover PS , and let U be a finite-dimensional (1) If T is a simple A-module then n dim HomA (PS , T ) = dim EndA (S) 0

a finite-dimensional algebra A A-module. if S ∼ = T, otherwise.

(2) The multiplicity of S as a composition factor of U is dim HomA (PS , U )/ dim EndA (S). (3) If e ∈ A is an idempotent then dim HomA (Ae, U ) = dim eU . Proof. (1) If PS → T is any non-zero homomorphism, the kernel must contain Rad PS , being a maximal submodule of S. Since PS / Rad PS ∼ = S is simple, the kernel must be ∼ Rad PS and S = T . Every homomorphism PS → S is the composite PS → PS / Rad PS → S of the quotient map and either an isomorphism of PS / Rad PS with S or the zero map. This gives an isomorphism HomA (PS , S) ∼ = EndA (S). (2) Let 0 = U0 ⊂ U1 ⊂ · · · ⊂ Un = U

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be a composition series of U . We prove the result by induction on the composition length n, the case n = 1 having just been established. Suppose n > 1 and that the multiplicity of S in Un−1 is dim HomA (PS , Un−1 )/ dim EndA (S). The exact sequence 0 → Un−1 → U → U/Un−1 → 0 gives rise to an exact sequence 0 → HomA (PS , Un−1 ) → HomA (PS , U ) → HomA (PS , U/Un−1 ) → 0 by 7.3, so that dim HomA (PS , U ) = dim HomA (PS , Un−1 ) + dim HomA (PS , U/Un−1 ). Dividing these dimensions by dim EndA (S) gives the result, by part (1). (3) There is an isomorphism of vector spaces HomA (Ae, U ) ∼ = eU specified by φ 7→ φ(e). Note here that since φ(e) = φ(ee) = eφ(e) we must have φ(e) ∈ eU . This mapping is injective since each A-module homomorphism φ : Ae → U is determined by its value on e as φ(ae) = aφ(e). It is surjective since the equation just written down does define a module homomorphism for each choice of φ(e) ∈ eU . Notice in 7.17 that if the field over which A is defined is algebraically closed then dim EndA (S) = 1, by Schur’s lemma, so that the multiplicity of S in U is just dim HomA (PS , U ). Again in the context of a finite-dimensional algebra A, we define for each pair of simple A-modules S and T the integer cST = the composition factor multiplicity of S in PT . These are called the Cartan invariants of A, and they form a matrix C = (cST ) with rows and columns indexed by the isomorphism types of simple A-modules, called the Cartan matrix of A. (7.18) COROLLARY. Let A be a finite-dimensional algebra over a field, let S and T be simple A-modules and let eS , eT be idempotents so that PS = AeS and PT = AeT are projective covers of S and T . Then cST = dim HomA (PS , PT )/ dim EndA (S) = dim eS AeT / dim EndA (S). While it is rather weak information just to know the composition factors of the projective modules, this is at least a start in describing these modules, and the information may be conveniently displayed as a matrix. We will see later on in the case of group algebras that there is an extremely effective way of computing the Cartan matrix using the decomposition matrix.

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Exercises for Section 7. 1. Let A be a finite-dimensional algebra over a field. Show that A is semisimple if and only if all finite-dimensional A-modules are projective. f g 2. Suppose that we have module homomorphisms U −→V −→W . Show that part of Proposition 7.6(a) can be strengthened to say the following: if gf is an essential epimorphism and f is an epimorphism then both f and g are essential epimorphisms. 3. In this question U, V and W are modules for a finite-dimensional algebra over a field. (a) Show that U → W is an essential epimorphism if and only if U is a homomorphic image of PW in such a way that the composite PW → U → W is the projective cover of W. (b) Prove the following ‘extension and converse’ to Nakayama’s lemma: let V be any submodule of U . Then U → U/V is an essential epimorphism ⇔ V ⊆ Rad U . 4. Let PS be an indecomposable projective module for a finite-dimensional algebra over a field. Show that every homomorphic image of PS (a) has a unique maximal submodule, and (b) is indecomposable. 5. Let A be a finite-dimensional algebra over a field, and suppose that f, f ′ are primitive idempotents of A. Show that the indecomposable projective modules Af and Af ′ are isomorphic if and only if f S = f ′ S for every simple module S. 6. Let A be a finite-dimensional algebra over a field, and suppose that Q is a projective A-module. Show that in any expression Q = PSn11 ⊕ · · · ⊕ PSnrr where S1 , . . . , Sr are non-isomorphic simple modules, we have ni = dim HomA (Q, Si )/ dim EndA (Si ). 7. Let A be a finite-dimensional algebra over a field. Suppose that V is an A-module, and that a certain simple A-module S occurs as a composition factor of V with multiplicity 1. Suppose that there exist non-zero homomorphisms S → V and V → S. Prove that S is a direct summand of V . 8. Let G = Sn , let k be a field of characteristic 2 and let Ω = {1, 2, . . . , n} permuted transitively by G. (a) When n = 3, show that the permutation module kΩ is semisimple, being the direct sum of the one-dimensional trivial module and the 2-dimensional simple module. (b) When n = 4 there is a normal subgroup V ⊳ S4 with S4 /V ∼ = S3 , where V = h(1, 2)(3, 4), (1, 3)(2, 4)i. The simple kS4 -modules are precisely the two simple kS3 -modules, made into kS4 -modules via the quotient homomorphism to S3 . Show that kΩ is uniserial with three composition factors which are the trivial module, the 2-dimensional simple module and the trivial module. [Use Exercise 18 from Section 6.]

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8. Projective modules for group algebras This section is a mix of general facts about group algebras and specific examples to do with certain types of group. We start with a summary of the situation for p-groups over a field of characteristic p, much of which we have already seen. (8.1) THEOREM. Let k be a field of characteristic p and G a p-group. The regular representation is an indecomposable projective module, which is the projective cover of the trivial representation. Every finitely generated projective module is free. The only idempotents in kG are 0 and 1. Proof. We have seen in 6.8 that Rad(kG) = IG and kG/ Rad(kG) ∼ = k. It follows immediately that kG is indecomposable, either directly by arguing that kG = U ⊕ V would imply k = U/ Rad U ⊕ V / Rad V , or using the theory of Section 7. By Nakayama’s lemma kG is the projective cover of k. By 7.13 and 6.3 every indecomposable projective is isomorphic to kG. Every finitely generated projective is a direct sum of indecomposable projectives, and so is free. Finally, every idempotent e ∈ kG gives a module decomposition kG = kGe ⊕ kG(1 − e). If e 6= 0 then we must have kG = kGe, so kG(1 − e) = 0 and e = 1. We can deduce from this some information about projective modules for arbitrary groups, using the following lemma, which is valid over an arbitrary commutative ring R. (8.2) LEMMA. Let H be a subgroup of G. (1) If P is a projective RG-module then P ↓G H is a projective RH-module. (2) If Q is a projective RH-module then Q ↑G H is a projective RG-module. Proof. (1) As a RH-module, RG ↓H ∼ =

M

RHg ∼ = (RH)|G:H|

g∈[H\G]

which is a free module. Hence a direct summand of RGn on restriction to H is a direct summand of RH |G:H|n , which is again projective. (2) We have H G∼ G∼ ∼ (RH) ↑G H = (R ↑1 ) ↑H = R ↑1 = RG so that direct summands of RH n induce to direct summands of RGn . (8.3) COROLLARY. Let k be a field of characteristic p and let pa be the exact power of p which divides |G|. If P is a projective kG-module then pa dim P .

Proof. Let H be a Sylow p-subgroup of G and P a projective kG-module. Then P ↓G H is projective by 8.2, hence free as a kH-module by 8.1, and of dimension a multiple of |H|.

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We now examine in detail the projective modules for several specific kinds of groups. The following result will be used in constructing these modules. It is valid over an arbitrary commutative ring R. (8.4) PROPOSITION. Suppose that V is any RG-module which is free as an Rmodule and P is a projective RG-module. Then V ⊗R P is projective. Proof. If P ⊕ P ′ ∼ = RGn then V ⊗ RGn ∼ = V ⊗ P ⊕ V ⊗ P ′ and it suffices to show that V ⊗ RGn is free. We offer two proofs of the fact that V ⊗ RG ∼ = RGrank V . The first G∼ G ∼ is that V ⊗ RG ∼ = V ⊗ (R ↑G 1 ) = (V ⊗ R) ↑1 = V ↑1 . As a module for the identity group, G∼ G rank V ∼ V is just a free R-module and so V ↑1 = (R ↑1 ) = RGrank V . The second proof is really the same as the first, but we make the isomorphism explicit. Let V triv be the same R-module as V , but with the trivial G-action, so V triv ∼ = Rrank V as RG-modules. We define a linear map V ⊗ RG → V triv ⊗ RG v ⊗ g 7→ g −1 v ⊗ g

,

which has inverse gw ⊗ g ← w ⊗ g. One checks that these mutually inverse linear maps are RG-module homomorphisms. Finally V triv ⊗ RG ∼ = RGrank V . In the calculations which follow we will need to use the fact that for representations over a field, taking the Kronecker product with a fixed representation preserves exactness. (8.5) LEMMA. Let 0 → U → V → W → 0 be a short exact sequence of kG-modules and X another kG-module, where k is a field. Then the sequence 0 → U ⊗k X → V ⊗k X → W ⊗k X → 0 is exact. Thus if U is a submodule of V then (V /U ) ⊗k X ∼ = (V ⊗k X)/(U ⊗k X). Proof. The exactness is a question of linear algebra which is independent of the group action. For the reader who knows a little homological algebra, the result is equivalent to the statement that higher Tor groups vanish over a field. We may also give a more direct approach by taking bases for the modules concerned. We may suppose that U is a submodule of V . Let v1 , . . . , vn be a basis for V such that v1 , . . . , vd is a basis for U and let x1 , . . . , xm be a basis for X. Now the vi ⊗k xj with 1 ≤ i ≤ n and 1 ≤ j ≤ m form a basis for V ⊗k X, and the same elements with 1 ≤ i ≤ d and 1 ≤ j ≤ m form a basis for U ⊗k X. This shows that U ⊗k X is a submodule of V ⊗k X, and the quotient has as a basis the images of the vi ⊗k xj with d + 1 ≤ i ≤ n and 1 ≤ j ≤ m, which is in bijection with a basis of W ⊗k X.

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(8.6) Example. We describe the projective kG-modules where k is a field of characteristic p and G = H × K where H is a p-group and K has order prime to p. Following this example we will consider projective modules for the semidirect products H ⋊ K and K ⋊H and the observations we will make when we do this also apply to the direct product. However, it seems profitable to consider the simpler example first. We make use of the following general isomorphism (not dependent on the particular hypotheses we have here): k[H × K] ∼ = kH ⊗ kK

as k-algebras,

which arises because kG has as a basis the elements (h, k) where h ∈ H, k ∈ K, and kH ⊗ kK has as a basis the corresponding elements h ⊗ k. These two bases multiply together in the same fashion, and so we have an algebra isomorphism. In this calculation the only tensor product which will appear is tensor product over the field k. We next observe that Rad kG = Rad kH ⊗ kK. This is because the quotient (kH ⊗ kK)/(Rad kH ⊗ kK) ∼ = k ⊗ kK = (kH/ Rad kH) ⊗ kK ∼ is semisimple, since kK is semisimple, so that Rad(kH ⊗ kK) ⊆ Rad kH ⊗ kK. On the other hand Rad kH ⊗ kK is a nilpotent ideal of kH ⊗ kK, so is contained in the radical, and we have equality. Let us write kK = S1n1 ⊕ · · · ⊕ Srnr , where S1 , . . . , Sr are the non-isomorphic simple kK-modules. Since H = Op (G), these are also the non-isomorphic simple kG-modules. We have kG = kH ⊗ kK = (kH ⊗ S1 )n1 ⊕ · · · ⊕ (kH ⊗ Sr )nr and so the kH ⊗ Si are projective kG-modules. Each occurs with multiplicity equal to the multiplicity of Si as a summand of kG/ Rad(kG), and so must be indecomposable, using 7.13. We have therefore constructed all the indecomposable projective kG-modules, and they are the modules PSi = kH ⊗ Si . Suppose that 0 ⊂ P1 ⊂ · · · ⊂ Pn = kH is a composition series of the regular representation of H. Since H is a p-group, all the composition factors are the trivial representation, k. Because ⊗k Si preserves exact sequences, the series 0 ⊂ P1 ⊗Si ⊂ · · · ⊂ Pn ⊗Si = PSi has quotients k ⊗ Si = Si , which are simple, and so this is a composition series of PSi . As an example, suppose that H is cyclic of order ps . Then kH = Ups has a composition series in which Pj = Rad(kH)j · kH ∼ = Uj and from the description of the radical of kG we see that the terms in the composition series of PSi are Pj ⊗ Si = Rad(kG)j · PSi . Because

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the radical series is in fact a composition series, it follows (as in Exercise 5 to Section 6) that this is the unique composition series of PSi , and that there are no more submodules of PSi other than the ones listed. We move on now to describe the projective kG-modules where k is a field of characteristic p and G = H ⋊ K where H is a p-group and |K| is prime to p. The last example is a special case of this situation, and so the results about to be obtained also apply in that case. However, our results will be less specific than in 8.6. Here, again, H = Op (G), and so the simple kG-modules are precisely the simple kK-modules. Since G acts on these via the ring surjection kG → kK, the kernel of this map acts as zero on all simple modules and so is contained in the radical. But also kK is a semisimple ring, so the kernel is the radical, and in particular it is a nilpotent ideal. (The kernel is in fact the ideal kG·IH considered in P 1 Exercise 10 of Section 6, but we do not need this here.) The idempotent e1 = |K| g∈K g is the primitive idempotent of kK which projects onto the unique 1-dimensional simple summand of kK. It is already an idempotent in kG, so it is its own lift to kG, and hence it is primitive in kG by 7.9. We conclude that Pk = kGe1 , by 7.12. This identifies Pk as a reasonably explicit subset of kG, but we now do better than this. (8.7) PROPOSITION. Let k be a field of characteristic p and let G = H ⋊ K where H is a p-group and K has order prime to p. Then Pk ∼ = kH where H acts on kH by left multiplication and K acts by conjugation. Proof. It comes as a surprise that there should be an action of G on kH in the manner P P P specified. Explicitly, if g∈H ag g, x ∈ H and y ∈ K then x g∈H ag g = g∈H ag xg and P P y g∈H ag g = g∈H ag ygy −1 . We could verify that this does give an action of G on kH, but instead we show that G acts on Pk via these formulae. We have kG = kH · kK and kKe1 = ke1 , so Pk ∼ = kGe1 = kHe1 . We claim that kHe1 ∼ = kH as k-vector spaces via an isomorphism in which he1 corresponds to h. Indeed the elements he1 are independent as h ranges through H, so we have just specified a bijection between two bases. Now H and K act on kHe1 as x · he1 = xhe1 for x ∈ H and y · he1 = yhy −1 ye1 = yhy −1 e1 for y ∈ K. Identifying kHe1 with kH, these actions are as specified in the statement of the result. We should point out that whenever we have a semidirect product G = H ⋊ K there is an action of G on kH in the way we have just described, but in general if H is not a Sylow p-subgroup of G this module will not be Pk . (8.8) PROPOSITION. Let k be a field of characteristic p and let G = H ⋊K where H is a p-group and K has order prime to p. If S is any simple kG-module then PS ∼ = Pk ⊗ S. ∼ There is an isomorphism of kG-modules Pk ⊗ kK = kG.

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Proof. By 8.4 Pk ⊗ S is projective. Tensoring the epimorphism Pk → k with S gives an epimorphism Pk ⊗ S → S. Now Rad(Pk ⊗ S) = Rad(kG) · (Pk ⊗ S) ⊇ Rad(kH) · (Pk ⊗ S) = (Rad(kH) · Pk ) ⊗ S since H is a normal p-subgroup and so acts trivially on S. Since (Rad(kH) · Pk ) ⊗ S has codimension in Pk ⊗ S equal to dim S we have that S ∼ = (Pk ⊗ S)/ Rad(Pk ⊗ S). This shows that Pk ⊗ S is the projective cover of S. Finally, since kK is identified as the radical quotient of kG, Pk ⊗ kK is the direct sum of projective modules PS , each appearing as often as S appears as a summand of kK. This description identifies Pk ⊗ kK as kG. Observe that since H is normal in G, IH is invariant under the conjugation action of K, and so Pk = kH ⊃ IH ⊃ IH 2 ⊃ · · · is a chain of kG-submodules of Pk . Our next aim is to describe the kG-module structure of the quotients IH s /IH s+1 , and for this we specialize to the case H = hxi ∼ = Cpn and K = hyi ∼ = Cq are cyclic groups with p6 q. Suppose that yx = xr , so that n G = hx, y xp = 1 = y q , yx = xr i.

In this case the powers IH s are a complete list of the submodules of Pk , and so it is a composition series of Pk as a kG-module. The action of y on IH is given by y(x − 1) = yx − 1 = xr − 1 = (x − 1)(xr−1 + · · · + x + 1 − r) + r(x − 1) ≡ r(x − 1) (mod IH 2 ). More generally for some α ∈ IH 2 , y(x − 1)s = (xr − 1)s = (r(x − 1) + α)s = r s (x − 1)s + sr(x − 1)s−1 α + · · · ≡ r s (x − 1)s

(mod IH s+1 ).

Thus y acts on the quotient IH s /IH s+1 as multiplication by r s . One way to describe this is that if W = IH/IH s then IH s /IH s+1 = W ⊗s , the s-fold tensor power. We now summarize these assertions.

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(8.9) PROPOSITION. Let k be a field of characteristic p and let G = hxi ⋊ hyi where hxi = Cpn and hyi = Cq has order prime to p. Suppose that yx = xr and let W be the 1-dimensional kG-module on which y acts as multiplication by r. If S is any simple kG-module then PS has a unique composition series, equal to the radical series, in which n the quotients Radi PS / Radi+1 PS are S, S ⊗ W, S ⊗ W ⊗2 , . . . , S ⊗ W ⊗p −1 = S. The assertion about the uniqueness of the composition series follows from Exercise 5 of Section 6, since the factors in the radical series are all simple. To continue this example we observe that the isomorphism types of the composition factors of PS occur in a cycle which repeats itself. Since x 7→ xr is an automorphism of Cpn there is a least positive integer f such that r f ≡ 1 (mod pn ). Put pn − 1 = ef . Then the modules k, W, W ⊗2 , W ⊗3 , . . . give rise to f different representations. They repeat e times in Pk , except for k which appears e + 1 times, and a similar repetition occurs with the composition factors S ⊗ W ⊗i of PS . We now consider the projective modules for groups which are a semidirect product of a p-group and a group of order prime to p, but with the roles of these groups the opposite of what they were in the last example. We say that a group G has a normal p-complement if and only if it has a normal subgroup K ⊳ G of order prime to p with |G : K| a power of p. Necessarily in this situation, if H is a Sylow p-subgroup of G then G = K ⋊ H by the Schur-Zassenhaus theorem. (8.10) THEOREM. Let G be a finite group and k a field of characteristic p. The following are equivalent. (1) G has a normal p-complement. (2) For every simple kG-module S, the composition factors of the projective cover PS are all isomorphic to S. (3) The composition factors of Pk are all isomorphic to k. Proof. (1) ⇒ (2): Let G = K ⋊H where p6 |K| and H is a Sylow p-subgroup of G. We show that kH, regarded as a kG-module via the homomorphism G → H, is a projective module. In fact, since kK is semisimple we may write kK = k ⊕ U for some kK-module G G G∼ U , and now kG = kK ↑G K = k ↑K ⊕U ↑K . Here k ↑K = kH as kG-modules (they are permutation modules with stabilizer K) and so kH is projective, being a summand of kG. Now if S is any simple kG-module then S ⊗k kH is also projective by 8.4, and all its composition factors are copies of S = S ⊗k k since the composition factors of kH are all k (using 6.3 and 8.5). The indecomposable summands of S ⊗k kH are all copies of PS , and their composition factors are all copies of S. (2) ⇒ (3) is immediate. (3) ⇒ (1): Suppose that the composition factors of Pk are all trivial. If g ∈ G is any ∼ t element of order prime to p (we say such an element is p-regular) then Pk ↓G hgi = k for

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some t, since khgi is semisimple. Thus g lies in the kernel of the action on Pk and if we put K = hg ∈ G g is p-regulari then K is a normal subgroup of G, G/K is a p-group and K acts trivially on Pk . We show that K contains no element of order p: if g ∈ K were such an element, then as Pk ↓G hgi is a projective khgi-module, it is isomorphic to a direct sum of copies of khgi by 8.1, and so g does not act trivially on Pk . It follows that p6 |K|, thus completing the proof.

We have already studied an example of the situation described in 8.10, namely kS3 where k is the field with four elements. In this example we may take H = h(1, 2)i. Observe that if V is the 2-dimensional simple kS3 -module then V ⊗ kH ∼ = V ⊕ V since V is projective, so that the module S ⊗ kH which appeared in the proof of 8.10 need not be indecomposable. Working over a field k, we next examine the behaviour of projective modules under the operation of dualization. We recall the definition U ∗ = Homk (U, k) and record the following properties:

(1) (2) (3) (4)

(8.11) PROPOSITION. Let k be a field. Then U ∗∗ ∼ = U as kG-modules, U is semisimple if and only if U ∗ is semisimple, U is indecomposable if and only if U ∗ is indecomposable, and a morphism f : U → V is a monomorphism (epimorphism) if and only if f ∗ : V ∗ → U ∗ is an epimorphism (monomorphism).

(8.12) PROPOSITION. Let k be a field. Then (1) kG∗ ∼ = kG as kG-modules, and (2) P is a projective kG-module if and only if P ∗ is a projective kG-module. Proof. (1) We denote the elements of kG∗ dual to the basis elements {g g ∈ G} by gˆ, so that gˆ(h) = δg,h ∈ k, the Kronecker δ. We define an isomorphism of vector spaces kG → kG∗ X X ag g 7→ ag gˆ .

g∈G

g∈G

To see that this is a kG-module homomorphism we observe that if x ∈ G then (xˆ g )(h) = gˆ(x−1 h) = δg,x−1 h = δxg,h = x cg(h)

for g, h ∈ G, so that xˆ g=x cg. ∗∗ ∼ (2) Since P = P it suffices to prove one implication. If P is a summand of kGn then P ∗ is a summand of (kGn )∗ ∼ = kGn , and so is also projective.

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When we previously introduced projective modules we could at the same time have defined injective modules, which enjoy properties similar to those of projective modules, but in a dual form. We say that a module I is injective if and only if whenever there are morphisms I x α  V

β

←− W

with β a monomorphism, then there exists a morphism γ : V → I so that γβ = α. Dually to Proposition 7.3, it is equivalent to require that every monomorphism I → V is split; and also that HomA ( , I) sends exact sequences to exact sequences. We leave this equivalence as an exercise. It is not true (in general) that injective modules are direct summands of free modules, but what we now show is that for group algebras over a field this special property does in fact hold. (8.13) COROLLARY. Let k be a field. (1) Projective kG-modules are the same as injective kG-modules. (2) Each indecomposable projective kG-module has a simple socle. Proof. (1) Suppose P is projective and that there are morphisms

V

β

P x α 

←− W

with β injective. Then in the diagram

V∗

β∗

P∗   ∗ yα

−→ W ∗

β ∗ is surjective, and so by projectivity of P ∗ there exists f : P ∗ → V ∗ such that β ∗ f = α∗ . Since f ∗ β = α we see that P is injective. To see that all injectives are projective, a similar argument shows that their duals are projective, hence injective, whence the original modules are projective, being the duals of injectives. (2) One way to proceed is to quote Exercise 6 of Section 6 which implies that Soc(P ) ∼ = ∗ ∗ ∗ ∗ ∗ (P / Rad P ) . If P is an indecomposable projective module then so is P and P / Rad P ∗ is simple. Thus so is Soc(P ).

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Alternatively, since homomorphisms S → P are in bijection (via duality) with homomorphisms P ∗ → S ∗ , if P is indecomposable projective and S is simple then P ∗ is also indecomposable projective and dim HomkG (S, P ) = dim HomkG (P ∗ , S ∗ ) n ∗ ∗ ∗ = dim End(S ) if P is the projective cover of S , 0 otherwise. Since dim End(S ∗ ) = dim End(S) this implies that P has a unique simple submodule and Soc(P ) is simple. An algebra for which injective modules and projective modules coincide is called selfinjective or quasi-Frobenius, so we have just shown that group rings of finite groups over a field are self-injective. For such an algebra, whenever a projective module occurs as a quotient of a submodule of another module, it is a direct summand. (8.14) COROLLARY. Suppose U is a kG-module, where k is a field, for which there are submodules U0 ⊆ U1 ⊆ U with U1 /U0 = P a projective module. Then U ∼ = P ⊕ U ′ for some submodule U ′ of U . Proof. The exact sequence 0 → U0 → U1 → P → 0 splits, and so U1 ∼ = P ⊕ U0 . Thus P is isomorphic to a submodule of U , and since P is injective the monomorphism P → U must split. We will now sharpen part (2) of 8.13 by showing that Soc PS ∼ = S, and we will also show that the Cartan matrix for group algebras is symmetric. These are properties which hold for a class of algebras called symmetric algebras, of which group algebras are examples. We say that a finite dimensional algebra A over a field k is a symmetric algebra if there is a non-degenerate bilinear form ( , ) : A × A → k such that (1) (symmetry) (a, b) = (b, a) for all a, b ∈ A, (2) (associativity) (ab, c) = (a, bc) for all a, b, c ∈ A. The group algebra kG is a symmetric algebra with the bilinear form defined on the basis elements by n 1 if gh=1 (g, h) = 0 otherwise as is readily verified. Notice that this bilinear form may be described on general elements a, b ∈ kG by (a, b) = coefficient of 1 in ab. For the record, a finite-dimensional algebra over a field on which there is a nondegenerate associative bilinear form is called a Frobenius algebra. It is the case that every Frobenius algebra is a quasi-Frobenius algebra, and it would have been possible to present a development leading to the results of Corollary 8.13 by proving this implication. We will use the bilinear form on kG in the proof of the next result, whose statement also holds for symmetric algebras in general.

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(8.15) THEOREM. Let P be an indecomposable projective module for a group algebra kG. Then P/ Rad P ∼ = Soc P . Proof. We may choose a primitive idempotent e ∈ kG so that P ∼ = kGe as kG-modules. We claim that Soc(kGe) = Soc(kG) · e, since Soc(kG) · e ⊆ Soc(kG) and Soc(kG) · e ⊆ kGe so Soc(kG) · e ⊆ kGe ∩ Soc(kG) = Soc(kGe), since the last intersection is the largest semisimple submodule of Ae. On the other hand Soc(kGe) ⊆ Soc(kG) since Soc(kGe) is semisimple so Soc(kGe) = Soc(kGe) · e ⊆ Soc(kG) · e. Next, Hom(kGe, Soc(kG)e) ∼ = e Soc(kG)e by 7.16, and since Soc(kG)e is simple, by 8.13, this is non-zero if and only if Soc(kG)e ∼ = kGe/ Rad(kGe). We show that e Soc(kG)e 6= 0. If e Soc(kG)e = 0 then 0 = (1, e Soc(kG)e) = (e, Soc(kG)e) = (Soc(kG)e, e) = (kG · Soc(kG)e, e) = (kG, Soc(kG)e · e) = (kG, Soc(kGe)). Since the bilinear form is non-degenerate this implies that Soc(kGe) = 0, a contradiction. (8.16) COROLLARY. Let k be a field. (1) If P is any projective kG-module and S is a simple kG-module, the multiplicity of S in P/ Rad P equals the multiplicity of S in Soc P . In particular dim P G = dim PG = dim(P ∗ )G = dim(P ∗ )G , where P G is the fixed points of G on P and PG denotes the largest trivial quotient of P. (2) For every simple kG-module S, (PS )∗ ∼ = PS ∗ . Proof. (1) This is true for every indecomposable projective module, hence also for every projective module. For the middle equality we may use an argument similar to the one which appeared in the proof of 8.13 (2). (2) We have seen in the proof of 8.13 that (PS )∗ is the projective cover of (Soc PS )∗ , and because of 8.15 we may identify the latter module as S ∗ .

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From this last observation we are able to deduce that, over a large enough field, the Cartan matrix of kG is symmetric. We recall that the Cartan invariants are the numbers cST = multiplicity of S as a composition factor of PT where S and T are simple. The precise condition we require on the size of the field is that it should be a splitting field, and this is something which is discussed in the next section. (8.17) THEOREM. Let k be a field and let S, T be simple kG-modules. The Cartan invariants satisfy cST · dim EndkG (T ) = cT S · dim EndkG (S). If dim EndkG (S) = 1 for all simple modules S (for example, if k is algebraically closed) then the Cartan matrix C = (cST ) is symmetric. Proof. We recall from 7.17 that cST = dim HomkG (PS , PT )/ dim EndkG (S) and in view of this we must show that dim HomkG (PS , PT ) = dim HomkG (PT , PS ). Now HomkG (PS , PT ) = Homk (PS , PT )G ∼ = (PS∗ ⊗k PT )G by 3.3 and 3.4. Since PS∗ ⊗k PT is projective by 8.4, this is the same as (PS∗ ⊗k PT )∗G ∼ = (PS ⊗k PT∗ )G ∼ = HomkG (PT , PS ), using 8.16. We conclude this section by summarizing some further aspects of injective modules. We define an essential monomorphism to be a monomorphism of modules f : V → U with the property that whenever g : U → W is a map such that gf is a monomorphism then g is a monomorphism. An injective hull (or injective envelope) of U is an essential monomorphism U → I where I is an injective module. By direct arguments, or by taking the corresponding results for essential epimorphisms and projective covers and applying the duality U 7→ U ∗ we may prove for finitely-generated kG-modules the following statements. • The inclusion Soc U → U is an essential monomorphism. g f • Given homomorphisms W →V →U , if two of f , g and f g are essential monomorphisms then so is the third. • A homomorphism f : V → U is an essential monomorphism if and only if f |Soc V : Soc V → Soc U is an isomorphism. • U → I is an injective hull if and only if I ∗ → U ∗ is a projective cover. Injective hulls always exist and are unique. From Theorem 8.15 we see that S → PS is the injective hull of the simple module S.

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• The multiplicity of a simple module S as a composition factor of a module U equals dim Hom(U, PS )/ dim End(S). Exercises for Section 8. 1. Prove that if G is any finite group then the only idempotents in the integral group ring ZG are 0 and 1. [If e is idempotent consider the rank of the free abelian group ZGe and also its image under the homomorphism ZG → Fp G for each prime p dividing |G|, which is a projective Fp G-module. Show that rankZ ZGe is divisible by |G|. Deduce from this that if e 6= 0 then e = 1.] 2. (a) Let H = C2 × C2 and let k be a field of characteristic 2. Show that (IH)2 is a P one-dimensional space spanned by h∈H h. (b) Let G = A4 = (C2 × C2 ) ⋊ C3 and let F4 be the field with four elements. Compute the radical series of each of the three indecomposable projectives for F4 A4 and identify each of the quotients Radn PS / Radn+1 PS . Now do the same for the socle series. Hence determine the Cartan matrix of F4 A4 . [Start by observing that F4 A4 has 3 simple modules, all of dimension 1, which one might denote by 1, ω and ω 2 . This exercise may be done by applying the kind of calculation which led to Proposition 8.9.] (c) Now consider F2 A4 where F2 is the field with two elements.  Prove that the 20 1 dimensional F2 -vector space on which a generator of C3 acts via is a simple 1 1 F2 C3 -module. Calculate the radical and socle series for each of the two indecomposable projective modules for F2 A4 and hence determine the Cartan matrix of F2 A4 . 3. Let G = H ⋊ K where H is a p-group, K is a p′ -group, and let k be a field of characteristic p. Regard kH as a kG-module via its isomorphism with Pk , so H acts as usual and K acts by conjugation. (a) Show that for each n, (IH)n is a kG-submodule of kH, and that (IH)n /(IH)n+1 is a kG-module on which H acts trivially. (b) Show that Pk = kH ⊇ IH ⊇ (IH)2 ⊇ (IH)3 · · · is the radical series of Pk as a kG-module. (c) Show that there is a map IH/(IH)2 ⊗k (IH)n /(IH)n+1 → (IH)n+1 /(IH)n+2 x + (IH)2 ⊗ y + (IH)n+1 7→ xy + (IH)n+2 which is a map of kG-modules. Deduce that (IH)n /(IH)n+1 is a homomorphic image of (IH/(IH)2)⊗n . (d) Show that the abelianization H/H ′ becomes a ZG-module under the action g · xH ′ =

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gxg −1 H ′ . Show that the isomorphism IH/(IH)2 → k ⊗Z H/H ′ specified by (x − 1) + (IH)2 7→ 1 ⊗ xH ′ of Section 6 Exercise 17 is an isomorphism of kG-modules. 4. The group SL(2, 3) is isomorphic to the semidirect product Q8 ⋊ C3 where the cyclic group C3 acts on Q8 = {±1, ±i, ±j, ±k} by cycling the three generators i, j and k. Assuming this structure, compute the radical series of each of the three indecomposable projectives for F4 SL(2, 3) and identify each of the quotients Radn PS / Radn+1 PS .

[Use Section 6 Exercise 15.] 5. Let G = P ⋊ S3 be a group which is the semidirect product of a 2-group P and the symmetric group of degree 3. (Examples of such groups are S4 = V ⋊ S3 where V = h(1, 2)(3, 4), (1, 3)(2, 4)i, and GL(2, 3) ∼ = Q8 ⋊ S3 where Q8 is the quaternion group of order 8.) (a) Let k be a field of characteristic 2. Show that kG has two non-isomorphic simple modules. (b) Let e1 , e2 , e3 ∈ F4 S3 be the orthogonal idempotents which appeared in Example 7.5. Show that each ei is primitive in F4 G and that dim F4 Gei = 2|P | for all i. [Use the fact that the F4 Gei are projective modules.] (c) Show that if e1 = () + (1, 2, 3) + (1, 3, 2) then F4 S4 e1 is the projective cover of the trivial module and that F4 S4 e2 and F4 S4 e3 are isomorphic, being copies of the projective cover of a 2-dimensional module. (d) Show that F4 Gei ∼ = F4 h(1, 2, 3)iei ↑G h(1,2,3)i for each i. 6. Let A be a finite-dimensional algebra over a field k, and let AA be the right regular representation of A. The vector space dual (AA )∗ = Homk (AA , k) becomes a left A-module via the action (af )(b) = f (ba) where a ∈ A, b ∈ AA and f ∈ (AA )∗ . Prove that the following two statements are equivalent: (a) (AA )∗ ∼ = A A as left A-modules. (b) There is a non-degenerate associative bilinear pairing A × A → k. An algebra satisfying these conditions is called a Frobenius algebra. Prove that, for a Frobenius algebra, projective and injective modules are the same thing. 7. Let A be a finite-dimensional algebra over a field k and suppose that the left regular representation A A is injective. Show that every projective module is injective and that every injective module is projective. 8. Let S and T be simple kG-modules, with projective covers PS and PT , where k is an algebraically closed field. (a) For each n prove that HomkG (PT , Socn PS ) = HomkG (PT / Radn PT , Socn PS ) = HomkG (PT / Radn PT , PS ).

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(b) Deduce Landrock’s theorem: the multiplicity of T in the nth socle layer of PS equals the multiplicity of S in the nth radical layer of PT . (c) Use Exercise 6 of Section 6 to show that these multiplicities equal the multiplicity of T ∗ in the nth radical layer of PS ∗ , and also the multiplicity of S ∗ in the nth socle layer of PT ∗ . 9. Let U be an indecomposable kG-module, where k is a field of characteristic p, and let Pk be the projective cover of the trivial module. Prove that dim((

X

g∈G

g) · U ) =

n

∼ Pk , 1 if U = 0 otherwise.

P For an arbitrary finite dimensional module V , show that dim(( g∈G g) · V ) is the multiplicity with which Pk occurs as a direct summand of V . P [Observe that kGG = PkG = k · g∈G g. Remember that Pk is injective and has socle isomorphic to k.] 10. Let U be a kG-module, where k is a field, and let PS be an indecomposable projective kG-module with simple quotient S. Show that in any decomposition of U as a direct sum of indecomposable modules, the multiplicity with which PS occurs is equal to dim HomkG (PS , U ) − dim HomkG (PS / Soc PS , U ) dim EndkG (S) and also to

dim HomkG (U, PS ) − dim HomkG (U, Rad PS ) . dim EndkG (S)

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9. Changing the ground ring: splitting fields and the decomposition map Suppose that A is an algebra A over a commutative ring R and that U is an A-module. If R → S is a homomorphism to another commutative ring S we may form the S-algebra S ⊗R A, and now S ⊗R U becomes an S ⊗R A-module in an evident way. In this section we study the relationship between U and S ⊗R U . When we specialize to a group algebra A = RG we will identify S ⊗R RG with SG.

We will pay special attention to two particular cases of this construction, the first being when R is a subring of S. If U is an A-module, we say that the module V = S ⊗R U is obtained from U by extending the scalars from R to S; and if an S ⊗R A-module V has the form S ⊗R U we say it can be written in R. In this situation, when U is free as an R-module we may identify U with the subset 1S ⊗R U of S ⊗R U , and an R-basis of U becomes an S-basis of S ⊗R U under this identification. In case A = RG is a group ring, with respect to such a basis of V = S ⊗R U the matrices which represent the action of elements g ∈ G on V have entries in R, and are the same as the matrices representating the action of these elements on U (with respect to the basis of U ). Equally, if we can find a basis for an SG-module V so that each g ∈ G acts by a matrix with elements in R then RG preserves the R-linear span of this basis, and this R-linear span is an RG-module U for which V = S ⊗R U . Thus an S-free module V can be written in R if and only if V has an S-basis with respect to which G acts via matrices with entries in R. The second situation to which we will pay particular attention arises when S = R/I for some ideal I in R. In this case applying S ⊗R to a module U is the same as reducing U modulo I. If V is an S ⊗R A-module of the form S ⊗R U for some A-module U we say that V can be lifted to U , and that U is a lift of V . Most often we will perform this construction when R is a local ring and I is the maximal ideal of R. We start by considering the behaviour of representations over a field. It is often a help to know that a representation can be written in a small field. (9.1) PROPOSITION. Let F ⊆ E be fields where E is algebraic over F and let A be a finite-dimensional F -algebra. Let V be a finite-dimensional E ⊗F A-module. Then there exists a field K with F ⊆ K ⊆ E, of finite degree over F , so that V can be written in K. Proof. Let a1 , . . . , an be a basis for A and let at act on V with matrix (atij ) with respect to some basis of V . Let K = F [atij , 1 ≤ t ≤ n, 1 ≤ i, j ≤ d]. Then [K : F ] is finite since K is an extension of F by finitely many algebraic elements, and A acts by matrices with entries in K.

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Let A be a finite-dimensional F -algebra, where F is a field. A simple A-module U is said to be absolutely simple if and only if E ⊗F U is a simple E ⊗F A-module for all extension fields E of F . We say that E is a splitting field for A if and only if every simple E ⊗F A-module is absolutely simple. If A is a group algebra, we say that E is a splitting field for G, by extension of the terminology. The kind of phenomenon which these definitions are designed to address is exemplified by cyclic groups. If G = hgi is cyclic of order n then g acts on CG as a direct sum of 2πi 1-dimensional eigenspaces with eigenvalues e n . Since these lie outside Q (if n ≥ 3), the regular representation of QG is a direct sum of simple modules but some of them have 2πi dimension greater than 1. On extending scalars to a field containing e n these simple modules decompose as direct sums of 1-dimensional modules. Thus Q is not a splitting 2πi field for G, but any field containing Q(e n ) is a splitting field since the simple modules are now 1-dimensional and remain simple on extension of scalars. We will use several times the fact that if we let F be any field, then F is a splitting field for the matrix algebra Mn (F ). In fact, if E ⊇ F is a field extension then E ⊗F Mn (F ) ∼ = Mn (E), an isomorphism which is most easily seen by observing that Mn (F ) has a basis consisting of the matrices Eij which are non-zero only in position (i, j), where the entry is 1. Thus E ⊗F Mn (F ) has a basis consisting of the elements 1 ⊗F Eij . Since these multiply together in the same fashion as the corresponding basis elements of Mn (E) we obtain the claimed isomorphism. Every simple Mn (F )-module is isomorphic to the module of column vectors of length n over F , and on extending the scalars to E we obtain column vectors of length n over E, which is a simple module for E ⊗F Mn (F ). As another example, the prime field Fp is a splitting field for every p-group, since the only simple Fp G-module here is Fp , which is absolutely simple. (9.2) PROPOSITION. Let U be a simple module for a finite-dimensional algebra A over a field F . The following are equivalent. (1) U is absolutely simple. (2) EndA (U ) = F . (3) The matrix algebra summand of A/ Rad A corresponding to U is Mn (F ), where n = dim U . Proof. (2) ⇒ (3): If the matrix summand of A/ Rad A corresponding to U is Mn (D) for some division ring D then D = EndA (U ). The hypothesis is that D = F so the matrix summand is Mn (F ) and since U identifies as column vectors of length n we have n = dim U . (3) ⇒ (1): The hypothesis is that A acts on U via a surjective ring homomorphism A → Mn (F ) where U is identified as F n . Now if E ⊇ F is an extension field then E ⊗ A acts on E ⊗ U = E n via the homomorphism E ⊗ A → E ⊗ Mn (F ) ∼ = Mn (E), which is n also surjective. Since E is a simple Mn (E)-module it follows that E ⊗F U is a simple E ⊗F A-module.

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(1) ⇒ (2): We prove this implication here only in the situation where F is a perfect field, so that all irreducible polynomials with coefficients in F are separable. The result is true in general and is not difficult but requires some technicality which we wish to avoid (see [CR]). This implication will not be needed for our application of the result. Suppose that EndA (U ) is larger than F , so there exists an endomorphism φ : U → U which is not scalar multiplication by an element of F . Let α be a root of the characteristic polynomial of φ in some field extension E ⊇ F : in other words, α is an eigenvalue of φ. Then 1E ⊗ φ : E ⊗F U → E ⊗F U is not scalar multiplication by α, because if it were the minimal polynomial of α over F would be a factor of (X − α)n where n = dimF U and by separability of the minimal polynomial we would deduce α ∈ F . Now 1E ⊗ φ − α ⊗ 1U ∈ EndE⊗A (E ⊗F U ) is a non-zero endomorphism with non-zero kernel, and since E ⊗F U is simple this cannot happen, by Schur’s lemma. (9.3) COROLLARY. Let A be a finite-dimensional algebra over a field F . Then A has a splitting field of finite degree over F . Proof. The algebraic closure F of F is a splitting field for A, since by Schur’s lemma condition (2) of Proposition 9.2 is satisfied for each simple F ⊗F A-module. By Proposition 9.1 there is a finite extension E ⊇ F so that every simple F ⊗F A-module can be written in E. The simple E ⊗F A-modules U which arise like this are absolutely simple, because if K ⊇ F is an extension field for which K ⊗F U is not simple then K ⊗F U is not simple, where K is an algebraic closure of K, and since K contains a copy of F , F ⊗F U cannot be simple since F is a splitting field for A, a contradiction. Finally, every simple E ⊗F A-module is isomorphic to one of the simple modules U which arise in this way. For, if V is a simple E ⊗F A-module let e2 = e ∈ E ⊗F A be an idempotent with the property that eV 6= 0 but eV 0 = 0 for all simple modules V 0 not isomorphic to V . Let W be a simple F ⊗F A-module which is not annihilated by e. We must have W ∼ = F ⊗F V since V is the only possible simple module which would give a result not annihilated by e. In the case of group algebras there is a finer result which was first conjectured by Schur and later proved by Brauer as a deduction from Brauer’s induction theorem. We state the result, but will not use it and do not prove it. The exponent of a group G is the least common multiple of the orders of its elements. (9.4) THEOREM (Brauer). Let G be a finite group, F a field, and suppose that F contains a primitive mth root of unity, where m is the exponent of G. Then F is a splitting field for G. 2πi

This theorem tells us that Q(e m ) and Fp (ζ) are splitting fields for G, where ζ is a primitive mth root of unity in an extension of Fp . Often smaller splitting fields than these

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can be found, and the determination of minimal splitting fields must be done on a caseby-case basis. For example, we may see as a result of the calculations we have performed earlier in this text that in every characteristic the prime field is a splitting field for S3 — the same is in fact true for all the symmetric groups. However, if we require that a field be 2πi a splitting field not only for G but also for all of its subgroups, then Q(e m ) and Fp (ζ) are the smallest possibilities, since as we have seen earlier a cyclic group of order n requires the presence of a primitive nth root of 1 in a splitting field. The reason for working with a splitting field is to get a complete picture of the composition factors of each module. When working with group representations in characteristic zero, if we did not take a splitting field the character table would not be square. When we do have a splitting field in characteristic zero, semisimplicity implies that knowing the behaviour of the simple modules tells us about all modules. In positive characteristic the situation is not so straightforward. Knowing that the simple modules do not break up further under extension of scalars implies the same thing for the indecomposable projective modules, and the Cartan matrix does not change. More specifically, if F is a splitting field for an algebra A and P is an indecomposable projective A-module then for every field extension E ⊇ F the E ⊗F A-module E ⊗F P remains indecomposable and projective, and if S is the simple quotient of P then E ⊗F S is the simple quotient of E ⊗F P . These statements appear in the exercises to this section. An indecomposable A-module U which remains indecomposable under all field extensions E ⊇ F is termed absolutely indecomposable. In general if the ground field F is not algebraically closed we can expect that there will be indecomposable A-modules which are not absolutely indecomposable, even though F may be a splitting field. We will not use it, but it is important to know that the statement of the following theorem is true. For a proof see [CRI, p. 139]. (9.5) THEOREM (Noether-Deuring). Let A be a finite-dimensional algebra over a field F and let E ⊇ F be a field extension. Suppose that U and V are A-modules for which E ⊗F U ∼ = E ⊗F V as E ⊗F A-modules. Then U ∼ = V as A-modules. Our next aim is to show that over a splitting field of characteristic p, the number of non-isomorphic simple representations of a group G equals the number of conjugacy classes of p-regular elements. Several proofs of this result are available, the first appearing in a paper of Brauer from 1932. The proof we shall present is also due to Brauer, coming from 1956. This proof is appealing because it is technically elementary, and could have appeared earlier in this text once we knew that the radical of a finite-dimensional algebra is nilpotent. We start with some lemmas. These have to do with a finite-dimensional algebra A over a field of characteristic p, and we will write S = linear span in A of {ab − ba a, b ∈ A} n T = {r ∈ A r p ∈ S for some n > 0}.

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(9.6) LEMMA. T is a linear subspace of A containing S. Proof. Since ab ≡ ba (mod S) we have (a + b)p ≡ ap + pap−1 b + · · · + bp ≡ ap + bp (mod S) n

m

and so if ap ∈ S, bp ∈ S with m ≥ n then m

m

m

m

m

(λa + µb)p ≡ λp ap + µp bp ≡ 0 (mod S). To show that S ⊆ T we show that (ab − ba)p ∈ S. Now (ab − ba)p ≡ (ab)p − (ba)p ≡ ap bp − bp ap ≡ 0 (mod S).

(9.7) LEMMA. If A = Mn (F ) is a matrix algebra where F is a field then S = T = matrices of trace zero. Proof. Since tr(ab − ba) = 0 we see that S is a subset of the matrices of trace zero. On the other hand when i 6= j every matrix Eij (zero everywhere except for a 1 in position (i, j)) can be written as a commutator: Eij = Eik Ekj − Ekj Eik , and also Eii − Ejj = Eij Eji − Eji Eij . Since these matrices span the matrices of trace zero we deduce that S consists exactly of the matrices of trace 0. Now S ⊆ T ⊆ A and S has codimension 1 so either T = S or T = A. The matrix E11 is idempotent and does not lie in T , so T = S. (9.8) PROPOSITION. Let A be a finite-dimensional algebra over a field of characteristic p which is a splitting field for A. The number of non-isomorphic simple representations of A equals the codimension of T in A. Proof. Let us write T (A), S(A), T (A/ Rad(A)), S(A/ Rad(A)) for the constructions S, T applied to A and A/ Rad(A). Since Rad(A) is nilpotent it is contained in T (A). Also (S(A) + Rad(A))/ Rad(A) = S(A/ Rad(A)) n

is easily verified. We claim that T (A)/ Rad(A) = T (A/ Rad(A)). For, if ap ∈ S then n (a + Rad(A))p ∈ S + Rad(A) and this shows that the left-hand side is contained in the n n right. Conversely, if (a + Rad(A))p ∈ S(A/ Rad(A)) then ap ∈ S(A) + Rad(A) ⊆ T (A) so T (A/ Rad(A)) ⊆ T (A)/ Rad(A). Now A/ Rad(A) is a direct sum of matrix algebras. It is apparent that both S and T preserve direct sums, so the codimension of T (A/ Rad(A)) in A/ Rad(A) equals the number of simple A-modules, and this equals the codimension of T (A) in A.

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Let p be a prime. An element in a finite group is said to be p-regular if it has order prime to p, and p-singular if it has order a power of p. The only element which is both p-regular and p-singular is the identity. (9.9) LEMMA. Let G be a finite group and p a prime. Each element x ∈ G can be uniquely written x = st where s is p-regular, t is p-singular and st = ts. If x1 = s1 t1 is such a decomposition of an element x1 which is conjugate to x then s is conjugate to s1 , and t is conjugate to t1 . Proof. If x has order n = αβ where α is a power of p and β is prime to p then we may take s = xα and t = xβ . If x = st = s1 t1 is a second such decomposition then s1 commutes with x and hence commutes with s and t which are powers of x. Similarly t1 commutes −1 −1 with s and t. Thus s−1 and now s−1 is p-singular, so 1 s = t1 t 1 s is p-regular and t1 t −1 these products equal 1, and s1 = s, t1 = t. If x1 = gxg then x1 = gsg −1 gtg −1 is a decomposition of x1 as a product of commuting p-regular and p-singular elements. Hence s1 = gsg −1 and t1 = gtg −1 by uniqueness of the decomposition. (9.10) LEMMA. Let F be a field and G a group. Then S is the set of elements of F G with the property that the sum of coefficients from each conjugacy class of G is zero. Proof. S is spanned by elements ab − ba with a, b ∈ G. Now ab − ba = a(ba)a −1 − ba is the difference of an element and its conjugate. Such elements exactly span the elements of F G which have coefficient sum zero on conjugacy classes. We come now to the result which is the goal of these lemmas. (9.11) THEOREM. Let F be a splitting field of characteristic p for a finite group G. The number of non-isomorphic simple F G modules equals the number of conjugacy classes of p-regular elements of G. Proof. We know that the number of simple F G modules equals the codimension of T in F G. We show this equals the number of p-regular conjugacy classes by showing that if x1 , . . . , xr is a set of representatives of the conjugacy classes of p-regular elements of G then x1 + T, . . . , xr + T is a basis of F G/T . If we write x = st where s is p-regular and t is p-singular, s and t commute, then n n n n n n (st − s)p = sp tp − sp = sp − sp = 0 for sufficiently large n, so that s + T = st + T . The elements g + T , g ∈ G do span F G/T , and now it follows from the last observation that we may throw out all except the p-regular elements and still have a spanning set. We show that the set which remains is linearly independent. P P P pn pn n Suppose that λi xi ∈ T . Then ( λi xi )p = λi xi ∈ S for sufficiently high pn n p , and there is always a sufficently large n so that xi = xi for all i, since the xi are P pn p-regular. Now λi xi ∈ S. But x1 , . . . , xr are independent modulo S by Lemma 9.10

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n

so λpi = 0 for all i, and hence λi = 0 for all i. This shows that x1 + T, . . . , xr + T are linearly independent.

Reduction modulo p and the decomposition map We turn now to the theory of reducing modules from characteristic zero to characteristic p, for some prime p, a theory developed principally by Richard Brauer. There is inherent interest in studying the relationships between representations in different characteristics, but apart from this we will obtain a remarkable way to compute the Cartan matrix of a group algebra, as well as a second proof of its symmetry. After that we use it in a study of characters whose degree is divisible by the order of a Sylow p-subgroup of G, and in the next section it will be used in a proof that the Cartan matrix is non-singular. There will be three rings in the set-up for reducing modules to characteristic p, and we list them as a triple (F, R, k). Here F is a field of characteristic zero equipped with a discrete valuation, R is the valuation ring in F with maximal ideal (π), and k = R/(π) is the residue field of R, which is required to have characteristic p. Such a triple is called a p-modular system. Given a finite group G, if both F and k are splitting fields for G we say that the triple is a splitting p-modular system for G. If we require that F contains a primitive mth root of unity (where m is the exponent of G) then necessarily R and k also contain primitive mth roots of unity and according to Brauer’s theorem both F and k are splitting fields. Without using Brauer’s theorem we may still deduce from 9.3 the existence of splitting p-modular systems where F is a number field (a finite extension of Q), and k is a finite field. We investigate some basic properties of representations of a finite group over a discrete valuation ring R. We comment that in this result and the next few results nothing specific to group representations is used, except that the group algebra of G is semisimple over a field of characteristic zero. The theory can be developed in the context of representations of an order in a finite-dimensional algebra, but this is a generality we do not pursue here.

and (1) (2) (3) (4)

(9.12) PROPOSITION. Let R be a discrete valuation ring with maximal ideal (π) residue field k = R/(π). Let G be a finite group. If S is a simple RG-module then πS = 0. The simple RG-modules are exactly the simple kG-modules made into RG-modules via the surjection RG → kG. For each RG-module U , πU ⊆ Rad(U ), and in particular (π)G ⊆ Rad(RG). For each RG-module U we have (Rad U )/πU = Rad(U/πU ).

Proof. (1) πS is an RG-submodule of S, so πS = S or 0. Since Rad R = (π) the R-module homomorphism S → S/πS is essential by Nakayama’s lemma, so that πS 6= S. Therefore πS = 0.

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(2) This follows from (1) since kG = RG/(π)G and (π)G annihilates the simple RGmodules. (3) This again follows from (1) since if V is a maximal submodule of U then U/V is simple so that πU ⊆ V , and it follows that πU is contained in all of the maximal submodules of U and hence in their intersection. (4) Rad U is the intersection of kernels of all the homomorphisms from U to simple modules. These homomorphisms all factor through the quotient homomorphism U → U/πU , and so Rad U is the preimage in U of the radical of U/πU , which is the statement we have to prove. (9.13) COROLLARY. Let R be a discrete valuation ring with maximal ideal (π) and residue field k = R/(π). Let G be a finite group. Let P and Q be projective RG-modules. Then P ∼ = Q as RG-modules if and only if P/πP ∼ = Q/πQ as kG-modules. Proof. If P/πP ∼ = Q/πQ as kG-modules then by 9.12 the radical quotients of P and Q are isomorphic, P/ Rad P ∼ = Q/ Rad Q. Now P and Q are projective covers of their radical quotients, by Nakayama’s lemma, so P ∼ = Q by uniqueness of projective covers. The converse implication is trivial. In the next pair of results we see that some important properties of idempotents and projective modules which we have already studied in the case of representations over a field, continue to hold when we work over a complete discrete valuation ring. It is a crucial hypothesis that the discrete valuation ring be complete. The idea of the proofs is the same as for the corresponding results over a field. (9.14) PROPOSITION. Let R be a complete discrete valuation ring with maximal ideal (π) and residue field k = R/(π). Let G be a finite group. Every expression 1 = e1 + · · · + en as a sum of orthogonal idempotents in kG can be lifted to an expression 1 = eˆ1 + · · · + eˆn in RG, where the eˆi ∈ RG are orthogonal idempotents with eˆi + (π) · RG = ei . Each idempotent ei is primitive if and only if its lift eˆi is primitive. Proof. The proof is very like the proofs of 7.10, 7.11 and 7.12. We start by showing simply that each idempotent e ∈ kG can be lifted to an idempotent eˆ ∈ RG. Consider the surjections of group rings (R/(π n ))G → (R/(π n−1 ))G for each n ≥ 2. Here (π n−1 )G/(π n )G is a nilpotent ideal in (R/(π n ))G and so by Theorem 7.9, any idempotent en−1 + (π n−1 )G ∈ (R/(π n−1 ))G can be lifted to an idempotent en + (π n )G ∈ (R/(π n ))G. Starting with an element e1 ∈ RG for which e1 + (π)G = e we obtain a sequence e1 , e2 , . . . of elements of RG which successively lift each other modulo increasing powers of (π), and so is a Cauchy sequence in RG. (The metric on RG comes from the valuation on R by taking the distance between two elements to be the maximum of the distances in the coordinate places.) This Cauchy sequence represents an element eˆ ∈ RG, since R is complete.

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Evidently eˆ is idempotent, because it is determined by its images modulo the powers of (π) and these are idempotent. It also lifts e. The argument that sums of orthogonal idempotents can be lifted now proceeds by analogy with the proof of 7.11, and the assertion that e is primitive if and only if eˆ is primitive is proved as in 7.12. (9.15) PROPOSITION. Let R be a complete discrete valuation ring with maximal ideal (π) and residue field k = R/(π). Let G be a finite group. (1) For each simple RG-module S there is an indecomposable projective RG-module PˆS = RGˆ eS with the property that PˆS /(π · PˆS ) ∼ = PS is the projective cover of S as a kGmodule. Here eˆS is a primitive idempotent in RG for which eˆS · S 6= 0. (2) The composite homomorphism PˆS → PS → S is a projective cover of S as an RGmodule. Furthermore, S is the unique simple quotient of PˆS . Thus PˆS ∼ = PˆT if and only if S ∼ = T. (3) Every finitely-generated RG module has a projective cover. (4) Every finitely-generated indecomposable projective RG-module is isomorphic to PˆS for some simple module S. Proof. (1) Let eS ∈ kG be a primitive idempotent for which eS · S 6= 0 and let eˆS ∈ RG be an idempotent which lifts eS , so that eˆS · S = e + S · S 6= 0. We define PˆS = RGˆ eS . Then PˆS is projective, and it is indecomposable since eˆS is primitive. Furthermore PˆS /(π · PˆS ) = kGeS , and defining this module to be PS it is a projective cover of S as a kG-module. (2) Now PˆS / Rad(PˆS ) ∼ = PS / Rad(PS ) by part (4) of 9.12, and this is isomorphic to S. Thus the composite epimorphism PˆS → PS → S is essential, by Nakayama’s lemma, and it is a projective cover. Since S is the radical quotient of PˆS it is the unique simple quotient of this module. This quotient determines the isomorphism type of PˆS by the uniqueness of projective covers. (3) Let U be a finitely-generated RG-module. Then U/ Rad U is a kG-module by 9.12, and it is semisimple, so U/ Rad U ∼ = S1 ⊕ · · · St for various simple modules Si . Consider the diagram PˆS1 ⊕ · · · ⊕ PˆSt   y U

−→

U/ Rad U

where the vertical arrow is the projective cover of S1 ⊕ · · · St as an RG-module. By projectivity we obtain a homomorphism PˆS1 ⊕ · · · ⊕ PˆSt → U which completes the triangle and it is an essential epimorphism by Proposition 7.7. Thus it is a projective cover. (4) Let P be a finitely-generated projective RG-module. By part (3) it has a projective cover, of the form α : PˆS1 ⊕ · · · ⊕ PˆSt → P . Since P is projective α must split, and there is a monomorphism β : P → PˆS1 ⊕ · · · ⊕ PˆSt with αβ = 1P . Since α is an essential

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epimorphism β must be an epimorphism also, so it is an isomorphism. If we suppose that P is indecomposable, then t = 1 and P ∼ = PˆS1 . Working over a principal ideal domain R, an RG-module L is called an RG-lattice if it is finitely-generated and free as an R-module. In more general contexts an RG-lattice is merely supposed to be projective as an R-module, but since projective modules are free over a principal ideal domain we do not need to phrase the definition that way here. We see, for example, that projective RG-modules are always lattices. Given an RG-lattice L (where R is a principal ideal domain with field of fractions F ) we may regard L as a subset of F ⊗R L. Conversely, if U is an F G-module, a full RG-lattice L in U is an RG-lattice L ⊆ U which has an R basis which is also an F -basis of U . In this situation U = F L ∼ = F ⊗R L. We say also that L is an R-form of U . We will show, after an intermediate lemma, that every finitely-generated F G-module contains a full RG-lattice, or in other words that every finitely-generated F G-module has an R-form. (9.16) LEMMA. Let R be a principal ideal domain with field of fractions F , and let U be a finite-dimensional F -vector space. Any finitely-generated R-submodule of U which contains an F -basis of U is a full lattice in U . Proof. Let L be a finitely-generated R-submodule of U which contains an F -basis of U . Since L is a finitely-generated torsion-free R-module, L ∼ = Rn for some n, and it has an R-basis x1 , . . . , xn . Since L contains an F -basis of U it follows that x1 , . . . , xn span U over F . We show that x1 , . . . , xn are independent over F . Suppose that λ1 x1 + · · · λn xn = 0 for certain λi ∈ F . We may write λi = abii where ai , bi ∈ R, since F is the field of fractions Q of R. Now clearing denominators we have ( bi )(λ1 x1 + · · · λn xn ) = 0 which implies that Q ( bi )λi = 0 for each i since x1 , . . . , xn is an R-basis. This implies that λi = 0 for all i and hence that n = d and x1 , . . . , xn is an F -basis of U . The kind of phenomenon which the last result is designed to exclude is exemplified by considering subgroups of R generated by elements which are independent over Q, such as √ the subgroup h1, 2i ∼ = Z2 . This is a free abelian group, but its basis is not an R-basis for R. According to the last lemma, such a phenomenon would not occur if R were the field of fractions of Z; indeed, the finitely-generated subgroups of Q are all cyclic. (9.17) PROPOSITION. Let (F, R, k) be a p-modular system and U a finite-dimensional F G-module. Then U may be written in R. Proof. Let u1 , . . . , un be any basis for U and let U0 be the R-submodule of U spanned by {gui i = 1, . . . , n, g ∈ G}. This is a finitely-generated R-submodule of U which contains an F -basis of U . Since R is a principal ideal domain with field of fractions F , by the last result U0 is a full RG-lattice in U , which is what we need to prove.

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The reason we introduce lattices here is that we can reduce them modulo ideals of R. Given an ideal I of R and an RG lattice L evidently V = L/(I ·L) is an (R/I)G-module. We

say that V is the reduction modulo I of the lattice L, and also that L is a lift from R/I to R

of V . Not every (R/I)G-module need be liftable from R/I to R. For example, taking R = Z and I = (p) with p ≥ 3, the group GL(2, p) has a faithful 2-dimensional representation over

Fp which cannot be lifted to Z, because such a representation would have to be faithful also, and on extending the scalars from Z to R would provide a 2-dimensional representation over R. The only finite subgroups of 2 × 2 real matrices are cyclic and dihedral. On the other hand the 2-dimensional faithful representation of GL(2, 2) over F2 does lift to Z.

The following result is crucial to the definition of the decomposition map, which will be given afterwards. (9.18) THEOREM (Brauer-Nesbitt). Let (F, R, k) be a p-modular system, G a finite group, and U a finitely-generated F G-module. Let L1 , L2 be full RG-lattices in U . Then L1 /πL1 and L2 /πL2 have the same composition factors with the same multiplicities, as kG-modules. Proof. We observe first that L1 + L2 is also a full RG-lattice in U , by Lemma 9.16, so by proving the result first for the pair of lattices L1 and L1 + L2 and then for L2 and L1 + L2 we see that it suffices to consider the case of a pair of lattices, one contained in the other. We now assume that L1 ⊆ L2 .

As R-modules, L1 and L2 are free of the same rank, and so L2 /L1 is a torsion module.

Hence L2 /L1 has a composition series as an R-module, and hence also as an RG-module, because every series of RG-modules can be extended to give a composition series of Rmodules. By working down the terms in a composition series, we see that it suffices to assume that L1 is a maximal RG-submodule of L2 , and we now make this assumption. Since L2 /L1 is a simple RG-module we have πL2 ⊆ L1 , by 9.12, and we consider the

chain of sublattices L2 ⊇ L1 ⊇ πL2 ⊇ πL1 . We must show that L2 /πL2 and L1 /πL1 have

the same composition factors. The composition factors of L1 /πL2 are common to both L2 /πL2 and L1 /πL1 , and we will complete the proof by showing that L2 /L1 ∼ = πL2 /πL1 . In fact, the map L2 → πL2 /πL1

x 7→ πx + πL1

is a surjection with kernel L1 .

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We should expect that much of the time when p |G| an F G-module U will contain non-isomorphic full sublattices. For example, if P is an indecomposable projective kGmodule, F ⊗R Pˆ is very often not a simple module. Writing F ⊗R P = S1 ⊕ · · · St as a sum of simple modules and taking a full RG-lattice in each Si , the direct sum of these lattices is not isomorphic to Pˆ , since Pˆ is indecomposable. More concretely, consider a cyclic group G = hgi of order 2 and let (F, R, k) be a 2modular system. The regular representation F G contains the full lattice R · 1 + R · g which is indecomposable since its reduction kG is indecomosable. It also contains the full lattice R(1 + g) + R(1 − g), which is a direct sum of RG-mouldes and hence is decomposable. We now define the decomposition matrix for a group G in characteristic p. Suppose that (F, R, k) is a splitting p-modular system for G. The decomposition matrix D is the matrix with rows indexed by the simple F G-modules and columns indexed by the simple kG-modules whose entries are the numbers dT S = multiplicity of S as a composition factor of L/πL where S is a simple kG-module, T is a simple F G-module and L is a full RG-lattice in T . We note that it is possible to make the definition of a decomposition matrix without the assumption that the p-modular system should be splitting, but this is not usual. Examples. 1. The decomposition matrices for S3 in characteristic 2 and characteristic 3 are:     1 0 1 0  1 0  and  0 1  . 0 1 1 1

2. When G is a p-group and k has characteristic p, the decomposition matrix has a single column and an entry for each ordinary simple character, that entry being the degree of the character. 3. The third example is sufficiently important that we state is a separate result. This is the situation where both F G-modules and kG-modules are semisimple.

(9.19) THEOREM. Let G be a group of order prime to p and let (F, R, k) be a p-modular system with R complete. With suitable ordering of the simple modules, the decomposition matrix is the identity matrix. In particular, F G and kG have the same number of simple modules, with the same dimensions. In fact more is true than this. Since the reduction modulo (π) of a tensor product is the tensor product of the reductions modulo (π), the tensor products of F G-modules and of kG-modules decompose in the same way, as do the inductions and restrictions of modules. Proof. Let T be a simple F G-module with full RG-lattice T0 . Then T0 /πT0 ∼ = S1 ⊕ · · ·⊕Sn for various simple kG-modules Si . Since these are projective, they lift to projective RG-lattices Sˆ1 , . . . , Sˆn which are the projective covers of S1 , . . . Sn . Thus the projective

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cover of T0 is a homomorphism Sˆ1 ⊕ · · · ⊕ Sˆn → T0 , and this is an isomorphism since the R-ranks of the two modules are the same. We deduce that T ∼ = F ⊗R Sˆ1 ⊕ · · · ⊕ F ⊗R Sˆn , and so n = 1 since T is simple. Thus every reduction of a simple module is simple. Equally, every simple kG-module is a composition factor of the reduction of some simple F G-module, since it is a composition factor of the reduction of F G, and so every simple kG-module does appear as the reduction of a simple F G-module. We state without proof at this point a particularly fine result about the decomposition map in the case of p-solvable groups. A group G is said to be p-solvable if it has a chain of subgroups 1 = G n / Gn / · · · / G 1 / G0 = G so that each factor Gi /Gi+1 is either a p-group or a group of order prime to p. (9.20) THEOREM (Fong, Swan, Rukolaine). Let (F, R, k) be a splitting p-modular system for a p-solvable group G. Then every simple kG-module is the reduction modulo (π) of an RG-lattice.

The cde triangle In order to express the decomposition matrix as the matrix of a linear map, we introduce three groups, which are instances of Grothendieck groups. Let (F, R, k) be a splitting p-modular system for a finite group G, and suppose that F and R are complete with respect to the valuation. We define G0 (F G) = free abelian group with the isomorphism types of simple F G-modules as a basis, G0 (kG) =free abelian group with the isomorphism types of simple kG-modules as a basis, K0 (kG) = free abelian group with the isomorphism types of indecomposable projective kG-modules as a basis. Thus G0 (F G) has rank equal to the number of conjugacy classes of G, and both G0 (kG) and K0 (kG) have rank equal to the number of p-regular conjugacy classes of G. If S is a simple F G-module we write [S] for the corresponding basis element of G0 (F G), similarly if S is a simple kG-module we write [S] for the corrsponding basis element of G0 (kG), and if P is an indecomposable projective kG-module we write [P ] for the corresponding basis element of K0 (kG). Extending this notation, if U is any kG-module with composition factors S1 , . . . , Sr occurring with multiplicities n1 , . . . , nr , we write [U ] = n1 [S1 ] + · · · + nr [Sr ] ∈ G0 (kG).

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There is a similar interpretation of [U ] ∈ G0 (F G) if U happens to be an F G-module, and if P = P1n1 ⊕ · · · ⊕ Prnr where the Pi are indecomposable projective kG-modules we put [P ] = n1 [P1 ] + · · · + nr [Pr ] ∈ K0 (kG). Since simple F G-modules biject with their characters, we may identify G0 (F G) with the subset of the space of class functions Ccc(G) consisting of the Z-linear combinations of the characters of the simple modules. Such Z-linear combinations of characters are termed virtual characters of G, so G0 (F G) is the group of virtual characters of F G. We now define the homomorphisms of the cde triangle, which are as follows: G0 (F G) e

d

% K0 (kG)

& c

−→

G0 (kG)

The definition of the homomorphism e on the basis elements of K0 (kG) is that if PS is an indecomposable projective kG-module then e([PS ]) = [F ⊗R PˆS ]. As observed in 9.13 and 9.15 the lift PˆS is unique up to isomorphism, and so this map is well-defined. The decomposition map d is defined thus on basis elements: if U is a simple F G-module containing a full RG-lattice L, we put d([U ]) = [L/πL]. By Theorem 9.18 this is welldefined, and in fact the formula works for arbitrary finite-dimensional F G-modules U , not just the simple ones. This definition means that the matrix of d is the transpose of the decomposition matrix. The homomorphism c is called the Cartan map and is defined by c([PS ]) = [PS ], where on the left the symbol [PS ] means the basis element of K0 (kG) corresponding to the indecomposable projective PS , and on the right [PS ] is an element of G0 (kG). From the definitions we see that the matrix of c is the Cartan matrix: P [PT ] = simple S cST [S] for each simple kG-module T . (9.21) PROPOSITION. c = de. Proof. It is simply a question of following through the definitions of these homomorphisms. If PS is an indecomposable projective kG-module then e[PS ] = [F ⊗R PˆS ]. To compute d[F ⊗R PˆS ] we choose any full RG-sublattice of F ⊗R PˆS and reduce it modulo (π). Taking PˆS to be that lattice, its reduction is PS and so de([PS ]) = [PS ] = c([PS ]). To investigate the properties of the cde triangle we study the relationship between homomorphisms between lattices and between their reductions modulo (π).

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(9.22) PROPOSITION. Let (F, R, k) be a p-modular system. Let U, V be F Gmodules containing full RG-lattices U0 and V0 . (1) HomRG (U0 , V0 ) is a full R-lattice in HomF G (U, V ). (2) π · HomRG (U0 , V0 ) = HomRG (U0 , π · V0 ) as a subset of HomRG (U0 , V0 )

(3) Suppose that U0 is a projective RG-lattice. Then

HomRG (U0 , V0 )/π · HomRG (U0 , V0 ) ∼ = HomRG (U0 , V0 /πV0 ) ∼ = HomkG (U0 /πU0 , V0 /πV0 ). Proof. (1) We should explain how it is that HomRG (U0 , V0 ) may be regarded as a subset of HomF G (U, V ). The most elementary approach is to take R-bases u1 , . . . , ur for U0 and v1 , . . . , vs for V0 . These are also F -bases for U and V . Any RG-hmomorphism U0 → V0 can be represented with respect to these bases by a matrix with entries in R. Regarding it as a matrix with entries in F , it represents an F G-module homomorphism U →V.

To see that HomRG (U0 , V0 ) is in fact a sublattice of HomF G (U, V ), we observe that HomRG (U0 , V0 ) ⊆ HomR (U0 , V0 ) ∼ = Rrs where r = dim U , s = dim V . The latter is a free R-module, so HomRG (U0 , V0 ) is an R-lattice since R is a principal ideal domain. We show it is full in HomF G (U, V ). Using the bases for U0 , V0 , let φ : U → V be an F G-module P homomorphism. Then φ(ui ) = λji vj with λji ∈ F . Choose a ∈ R so that aλji ∈ R for all i, j. Then aφ : U0 → V0 , showing that φ belongs to F · HomRG (U0 , V0 ). Therefore HomRG (U0 , V0 ) spans HomF G (U, V ) over F . (2) The map V0 → π · V0 given by x 7→ πx is an RG-isomorphism so the morphisms π U0 → π · V0 are precisely those which arise as composites U0 → V0 →πV0 , which are in turn the elements of π · HomRG (U0 , V0 ). (3) Consider Hom(U0 , V0 ) → Hom(U0 , V0 /πV0 ). Its kernel is Hom(U0 , πV0 ), which equals π Hom(U0 , V0 ). Since U0 is projective, the map is surjective, and it gives rise to the first isomorphism. For the second, all homomorphisms α : U0 → V0 /πV0 contain πU0 β

in the kernel, and so factor as U0 → U0 /πU0 →V0 /πV0 . The correspondence of α and β provides the isomorphism.

(9.23) COROLLARY. Suppose U0 and V0 are full RG-lattices in U and V , and U0 is projective. Then dimF HomF G (U, V ) = dimk HomkG (U0 /πU0 , V0 /πV0 ). Proof. Both sides equal rankR HomRG (U0 , V0 ) by parts (1) and (3) of the last result.

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(9.24) THEOREM. Let (F, R, k) be a splitting p-modular system for G and suppose that R is complete with respect to its valuation. Let S be a simple kG-module and let T be a simple F G-module containing a full RG-lattice T0 . The multiplicity of T in F ⊗R PˆS equals the multiplicity of S as a composition factor of T0 /πT0 . Proof. Applying the last corollary to the full RG-lattice PˆS of F ⊗R PˆS we obtain dimF HomkG (F ⊗R PˆS , T ) = dimk HomkG (PS , T0 /πT0 ), which shows that the multiplicities are equal. We comment that this equality of dimensions gives a second proof of the BrauerNesbitt theorem that the decomposition numbers dT S = dim HomkG (PS , T0 /πT0 ) are defined independently of the choice of lattice T0 , since the left-hand side in the equality does not depend on this choice. (9.25) COROLLARY.Let (F, R, k) be a splitting p-modular system for G and suppose that R is complete with respect to its valuation. With respect to the bases of G0 (F G), G0 (kG) and K0 (kG) by which these groups were defined (namely the bases whose elements are the symbols [T ], [S] and [PS ] where T is a simple F G-module and S is a simple kGmodule) the matrix of e is D and the matrix of d is D T , where D is the decomposition matrix. Proof. We have already observed that the matrix of d is D T . The entries eT S in the matrix of e are defined by e([PS ]) = [F ⊗R PˆS ] =

X

eT S T,

T

so that eT S is the multiplicity of T in F ⊗R PˆS . By the last result, eT S = dT S . Example. This result allows us to compute the characters of the indecomposable projective RG-modules PˆS (or more properly the characters of the F G-modules F ⊗R PˆS ). Using the decomposition matrices for S3 which were previously computed we see that in characteristic 2, χPˆ1 = χ1 + χ χPˆ2 = χ2 , and in characteristic 3

χPˆ1 = χ1 + χ2 χPˆ = χ + χ2 .

We now have a second proof of the symmetry of the Cartan matrix, but perhaps more importantly an extremely good way to calculate it. The effectiveness of this approach will be increased once we know about Brauer characters, which are treated in the next section.

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(9.26) COROLLARY. Let (F, R, k) be a splitting p-modular system for G. Then the Cartan matrix C = D T D. Thus C is symmetric. Example. When G = S3 the Cartan matrices in characteristic 2 and in characteristic 3 are 

1 0

1 0 0 1





  1 0 1 0 = 2 0 0 1

0 1



and



1 0

0 1 1 1





1 0 1

  0 2 1 = 1 1

1 2



as we have seen already. The equality of dimensions which played the key role in the proof of Theorem 9.24 can be nicely expressed in terms of certain bilinear pairings between the various Grothendieck groups. On the vector space of class functions on G we already have defined a Hermitian form and on the subgroup G0 (F G) it restricts to give a bilinear form h ,

i : G0 (F G) × G0 (F G) → Z

specified by h[U ], [V ]i = dim HomF G (U, V ) when U and V are F G-modules. We also have a pairing h ,

i : K0 (F G) × G0 (kG) → Z

specified by h[P ], [V ]i = dim HomkG (P, V ) when P is a projective kG-module and V is a kG-module. By Proposition 7.17 this quantity depends only on the composition factors of V , not on the actual module V , and so this pairing is well-defined. We readily see that each of these bilinear pairings is non-degenerate, in that in each case the free abelian groups have bases which are dual to each other. Thus if U and V are simple F G-modules we have h[U ], [V ]i = δ[U ],[V ] , and if S and T are simple kG-modules we have h[PS ], [T ]i = δ[S],[V ] . The equality of dimensions which appeared in the proof of 9.25 can now be expressed as follows. If x ∈ K0 (kG) and y ∈ G0 (F G) then he(x), yi = hx, d(y)i. This formalism is an expression of the fact that e and d are the transpose of each other.

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Blocks of defect zero We will use the relationship between representations in characteristic zero and in characteristic p to show that simple representations in characteristic zero whose degree is divisible by the order of a Sylow p-subgroup of G biject with simple representations in characteristic p which are projective. We study blocks more fully in a later section, and it turns out that representations with the special properties just described correspond to blocks of defect zero. We will deduce that each character of degree divisible by the order of a Sylow p-subgroup vanishes on elements of order divisible by p. (9.27) PROPOSITION. Let (F, R, k) be a p-modular system and G a finite group. Let Pˆ be a projective RG-module and χ the character of F ⊗R Pˆ . Then χ(1) is divisible by the order of a Sylow p-subgroup of G, and if g ∈ G has order divisible by p then χ(g) = 0. Proof. Since Pˆ /π Pˆ is a projective kG-module it has dimension divisible by the order of a Sylow p-subgroup of G, and this dimension equals the rank of Pˆ and also dim F ⊗R Pˆ , which equals χ(1). Consider now an element g ∈ G of order divisible by p. To show that χ(g) = 0 it suffices to consider Pˆ as an Rhgi-module, and as such it is still projective. We may suppose that Pˆ is an indecomposable projective Rhgi-module. Let us write g = st where s is p-regular, t is p-singular and st = ts, as in Lemma 9.9, so hgi = hsi × hti. As in Example 8.6 we can write Pˆ /π Pˆ = S ⊗ khti where S is a simple khsimodule. Since khsi is semisimple and S is projective as a khsi-module, we can lift S to a ˆ Sˆ = S. Now Pˆ /π Pˆ = S⊗khti = S⊗k ↑hgi = S ↑hgi . projective Rhsi-module Sˆ for which S/π hsi hsi hgi ˆ ˆ This lifts to S ↑ , which is a projective Rhgi-module. It is the projective cover of P /π Pˆ , hsi

hgi hgi so by uniqueness of projective covers Pˆ ∼ = Sˆ ↑hsi . We deduce that χ = χSˆ ↑hsi . It follows that χ(g) = 0 from the formula for an induced character, since no conjugate of g lies in hsi.

(9.28) THEOREM. Let (F, R, k) be a splitting p-modular system in which R is complete, and let G be a group of order pd q where q is prime to p. Let T be an F G-module of dimension n, containing a full RG-sublattice T0 . The following are equivalent. (1) pd n and T is a simple F G-module. (2) The homomorphism RG → EndR (T0 ) which gives the action of RG on T0 identifies EndR (T0 ) ∼ = Mn (R) with a ring direct summand of RG. (3) T is a simple F G-module and T0 is a projective RG-module. (4) The homomorphism kG → Endk (T0 /πT0 ) identifies Endk (T0 /πT0 ) ∼ = Mn (k) with a ring direct summand of kG.

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(5) As a kG-module, T0 /πT0 is simple and projective. Proof. (1) ⇒ (2) Suppose that (1) holds. We will use the formula obtained in Theorem 3.23 for the primitive central idempotent e associated to T , namely e=

n X χT (g −1 )g |G| g∈G

where χT is the character of T . The homomorphism ρ : F G → EndF (T ) which expresses the action of G on T identifies eF G with the matrix algebra End F (T ), and has kernel (1 − e)F G. Because pd n and the values of χT are sums of roots of unity we have e ∈ RG so that RG = eRG ⊕ (1 − e)RG as a sum of rings. Consider the homomorphism RG → EndR (T0 ) which expresses the action of RG on T0 . It is the restriction of ρ to RG, which takes values in EndR (T0 ), so has kernel (1 − e)F G ∩ RG = (1 − e)RG. We will show that this homomorphism is surjective. From this it will follow that eRG ∼ = EndR (T0 ) ∼ = Mn (R), a direct summand of RG. As an extension of the formula in Theorem 3.23 for the primitive central idempotent corresponding to T , we claim that if φ ∈ EndF (T ) then φ=

n X tr(ρ(g −1 )φ)ρ(g). |G| g∈G

To demonstrate this it suffices to consider the case φ = ρ(h) where h ∈ G, since these elements span EndF (T ). In this case n X n X tr(ρ(g −1 )ρ(h))ρ(g) = ρ(h) tr(ρ(g −1 h))ρ(h−1 g) |G| |G| g∈G

g∈G

= ρ(h)ρ(e) = ρ(h)

using the previously obtained formula for e. Now if φ ∈ EndR (T0 ), which we regard as P n −1 a subset of EndF (T ), we have |G| )φ)g ∈ RG, which shows that RG → g∈G tr(ρ(g EndR (T0 ) is surjective. This completes the proof of this implication. (2) ⇒ (3) Certainly T0 is projective as a module for EndR (T0 ) since it identifies with the module of column vectors for this matrix algebra. Assuming (2), we have that T 0 is a projective RG-module, since RG acts via its summand eRG which identifies with the matrix algebra. Furthermore, F G acts on T ∼ = F ⊗R T0 as column vectors for a matrix algebra over F , so T is a simple F G-module. (2) ⇒ (4) The decomposition RG = eRG⊕(1−e)RG with eRG ∼ = Mn (R) is preserved on reducing modulo (π), and we obtain kG = e¯kG ⊕ (1 − e¯)kG where e¯ is the image e in kG. Here e¯kG ∼ = Mn (k) and the action of kG on T0 /πT0 is via projection onto e¯kG.

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(3) ⇒ (5) Since T0 is a direct summand of a free RG-module it follows that T0 /πT0 is a direct summand of a free kG-module, and hence is projective. Furthermore, we claim that T0 /πT0 is indecomposable. For writing T0 /πT0 = PS1 ⊕ · · · ⊕ PSt where the PSi are indecomposable projectives we have that T0 /πT0 is the projective cover of S1 ⊕ · · · ⊕ St as an RG-module. It follows that T0 ∼ = PˆS1 ⊕ · · · ⊕ PˆSt and T ∼ = F ⊗R PˆS1 ⊕ · · · ⊕ F ⊗R PˆSt , so t = 1 since T is simple. By Theorem 9.24 the column of the decomposition matrix corresponding to S1 consists of zeros except for an entry 1 in the row of T . Since C = D T D, the multiplicity of S1 as a composition factor of PS1 is 1. We know from Theorem 8.15 that Soc PS1 ∼ = S1 and also PS1 / Rad PS1 ∼ = S1 , so that S1 occurs as a composition factor of PS1 with multiplicity at least 2 unless PS1 = S1 . This shows that T0 /πT0 is simple. (4) ⇒ (5) This is analogous to the proof (2) ⇒ (3). (5) ⇒ (1) Since T0 /πT0 is projective its dimension is divisible by pd by Corollary 8.3, and this dimension equals rankR T0 = dim T . If T were not simple as an F G-module we would be able to write T = U ⊕ V , and taking full RG-lattices U0 , V0 the composition factors of T0 /πT0 are the same as those of U0 /πU0 ⊕ V0 /πV0 . Since T0 /πT0 is in fact simple, this situation cannot occur, and T is simple. Statement (5) of Theorem 9.28 appears to depend on the F G-module T , but this is not really the case. If P is any simple projective kG-module, it can be lifted to an RGmodule Pˆ and taking T = F ⊗R Pˆ we have a module with a full RG-lattice T0 for which T0 /πT0 ∼ = P . Thus every simple projective kG-module is acted on by kG via projection onto a matrix algebra direct summand of kG, and the module T just constructed is always simple. Notice in the statement of Theorem 9.28 that since the full RG-sublattice T0 is arbitrary, every full RG-sublattice of T is projective. Since such a full lattice T0 is the projective cover of T0 /πT0 , which is a simple module defined independently of the choice of T0 , all such lattices T0 are isomorphic as RG-modules. Looking at the various equivalent statements of Theorem 9.28, one might be led to suspect that if k is a field of characteristic p and S is a simple kG-module of dimension divisible by the largest power of p which divides |G| then S is necessarily projective, but in fact this conclusion does not always hold. (9.29) COROLLARY. Let T be a simple F G-module, where F is a splitting field for G of characteristic 0, and let χT be the character of T . Let p be a prime, and suppose that the highest power of p which divides |G| also divides the degree χT (1). Then if g is any element of G of order divisible by p we have χT (g) = 0. Proof. This combines Proposition 9.27 with Theorem 9.28. If F does not initially appear as part of a p-modular system, we may replace F by a subfield which is a splitting field and which is a finite extension of Q, since QG has a splitting field of this form and by the argument of 9.3 it may be chosen to be a subfield of F . We may write T in this subfield without changing its character χT and the hypothesis about the power of p which

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divides χT (1) remains the same. Take the valuation on F determined by a maximal ideal p of the ring of integers for which p ∩ Z = (p), and complete F with respect to this valuation to get a splitting, complete, p-modular system. We may now apply Theorem 9.28. The above corollary is a significant piece of information about the character table of a group, which can be observed in many examples. Thus the simple character of degree 2 of the symmetric group S3 is zero except on the 2-regular elements, and the simple characters of S4 of degree 3 are zero except on the 3-regular elements. It is notable that in order to prove this result, which is stated within a characteristic zero framework, we have used technical machinery from characteristic p. Exercises for Section 9. 1. Let E = Fp (t) be a transcendental extension of the field with p elements and let F be the subfield Fp (tp ). Write α = tp ∈ F , so that tp − α = 0. Let A = E, regarded as an F -algebra. (a) Show that A has a simple module which is not absolutely simple. (b) Show that E is a splitting field for A, and that the regular representation of E ⊗F A is a uniserial module. Show that Rad(E ⊗F A) 6= E ⊗F Rad(A). [Notice that E ⊗F Rad(A) is always contained in the radical of E ⊗F A when A is a finite-dimensional algebra, being a nilpotent ideal.] (c) Show that A is not isomorphic to F G for any group G. 2. Let G be a cyclic group, F a field and S a simple F G-module. Show that E = EndF G (S) is a field with the property that E ⊗F S is a direct sum of modules which are all absolutely simple. 3. Let A be a finite-dimensional algebra over a field F and suppose that F is a splitting field for A. Let E ⊇ F be a field extension. Prove that Rad(E ⊗F A) ∼ = E ⊗F Rad(A). [The observation at the end of question 1 may help here.] 4. Let A be an F -algebra where F is a splitting field for A, and let E ⊇ F be a field extension. Show that every simple E ⊗F A-module can be written in F . [Bear in mind the result of question 3.] 5. Let A be a finite-dimensional F -algebra and let E ⊇ F be a field extension where E is a splitting field for A. Suppose that every simple A-module remains simple on extending scalars to E. Show that F is a splitting field for A. 6. Let A be a finite-dimensional F -algebra and E ⊇ F a field extension. (a) Show that if U → V is an essential epimorphism of A-modules then E ⊗F U → E ⊗F V is an essential epimorphism of E ⊗F A-modules. (b) Show that if P → U is a projective cover then so is E ⊗F P → E ⊗F U . 7. Let A be a finite-dimensional F -algebra where F is a splitting field for A. Let P be an indecomposable projective A-module. Show that if E ⊇ F is any field extension then E ⊗F P is indecomposable and projective as an E ⊗F A-module. Show further that every indecomposable projective E ⊗F A-module can be written in F .

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8. Let G = C2 × C2 be generated by elements a and b, and let E be a field of characteristic 2. Let t ∈ E be any element, which may be algebraic or transcendental over F2 . Let ρ : G → GL(E 2 ) be the representation with ρ(a) =



1 0

1 1



,

ρ(b) =



1 t 0 1



.

Show that this representation is absolutely indecomposable, and that it cannot be written in any proper subfield of F2 (t). 9. Let (F, R, k) be a p-modular system and suppose that R is complete. Let L and M be RG-lattices, where G is a finite group. (a) Show that L is a projective RG-module if and only if L/πL is a projective kGmodule. [Consider the projective cover of L.] (b) Deduce that if L/πL ∼ = M/πM as kG-modules and that L/πL is a projective kG∼ module then L = M as RG-modules. In other words, projective kG-modules lift uniquely to RG-lattices. 10. Let (F, R, k) be a p-modular system and G a finite group. Show that if U = U1 ⊕U2 is a finite-dimensional F G-module and L is a full RG-lattice in U then L ∩ U1 , L ∩ U2 are full RG-lattices in U1 and U2 , but that it need not be true that L = (L ∩ U1 ) ⊕ (L ∩ U2 ). [Consider the regular representation when G = C2 .] 11. Let U be the 2-dimensional representation of S3 over Q which is defined by requiring that with respect to a basis u1 , u2 the elements (1, 2, 3) and (1, 2) act by matrices 

0 −1 1 −1



and



1 0

−1 −1



.

Let U0 be the ZS3 -lattice which is the Z-span of u1 and u2 in U . Show that U0 /3U0 has just 3 submodules as a module for (Z/3Z)S3 , namely 0, the whole space, and a 1-dimensional submodule. Deduce that U0 /3U0 is not semisimple. Now let U1 be the Z-span of the vectors 2u1 +u2 and −u1 +u2 in U . Show (for example, by drawing a picture in which the angle between u1 and u2 is 120◦ , or else algebraically) that U1 is a ZS3 -lattice in U , and that it has index 3 in U0 . Write down matrices which give the action of (1, 2, 3) and (1, 2) on U1 with respect to the new basis. Show that U1 /3U1 also has just 3 submodules as a (Z/3Z)S3 -module, but that it is not isomorphic to U0 /3U0 . Identify U0 /U1 as a (Z/3Z)S3 -module. Prove that U1 is the unique ZS3 -sublattice of U0 of index 3. Show that Z3 ⊗S U0 is a uniserial Z3 S3 -lattice, where Z3 denotes the 3-adic integers. 12. Let (F, R, k) be a splitting p-modular system and G a finite group. Let T be an F G-module with the property that every full RG-sublattice of T is indecomposable and n n n+1 projective. Show that T is simple of degree divisible by p , where p |G|, p 6 |G|.

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10. Brauer characters To define Brauer characters for a finite group G we start with a p-modular system (F, R, k) and assume that both F and k contain a primitive ath root of unity, where a is the l.c.m. of the orders of the p-regular elements of G. If we wish to define the Brauer character of a kG-module U where k or F do not contain a primitive ath root of unity, we first extend the scalars so that the p-modular system does have this property. Let us put µF = {ath roots of 1 in F }

µk = {ath roots of 1 in k.}

The polynomial X a − 1 is separable both in F [X] and k[X] since its formal derivative d a a−1 is not zero and has no factors in common with X a − 1, so both dX (X − 1) = aX µF and µk are cyclic groups of order a. Also µF ⊆ R since roots of unity have value 1 under the valuation. We claim that the quotient homomorphism R → R/(π) = k gives ˆ → λ. This is because X a − 1 reduces to the an isomorphism µF → µk , which we write λ polynomial which is written the same way, and so its linear factors over F must map to ˆ over F the complete set of linear factors over k. These linear factors have the form X − λ and X − λ over k, so we obtain a bijection between the two groups of roots of unity. Let g ∈ G be a p-regular element (an element of order prime to p), and let ρ : G → GL(U ) be a representation over k. Then ρ(g) is diagonalizable, since khgi is semisimple and all eigenvalues of ρ(g) lie in k, being ath roots of unity. If the eigenvalues of ρ(g) are λ1 , . . . , λn we put ˆ1 + · · · + λ ˆn, φU (g) = λ and this is the Brauer character of U . It is a function which is only defined on the p-regular elements of G, and takes values in a field of characteristic zero, which we may always take to be C.   0 1 Example. Working over F2 , the specification ρ(g) = provides a 2-dimensional 1 1 representation U of the cyclic group hgi of order 3. The characteristic polynomial of this matrix is t2 + t + 1 and its eigenvalues are the primitive cube roots of unity in F4 . These lift to primitive cube roots of unity in C, and so φU (g) = e2πi/3 + e4πi/3 = −1. It is very tempting in this situation to observe that the trace of ρ(g) is 1, which can be lifted to 1 ∈ C, and to deduce that φU (g) = 1; however, this deduction is incorrect. We list the immediate properties of Brauer characters.

and (1) (2) (3)

(10.1) PROPOSITION. Let (F, R, k) be a p-modular system, let G be a finite group, let U , V , S be finite-dimensional kG-modules. φU (1) = dimk U . φU is a class function on p-regular conjugacy classes. φU (g −1 ) = φU (g) = φU ∗ (g).

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(4) φU ⊗V = φU · φV . (5) If 0 → U → V → W → 0 is a short exact sequence of kG-modules then φV = φU +φW . In particular, φU depends only on the isomorphism type of U . Furthermore, if U has P composition factors S, each occuring with multiplicity nS , then φU = S nS φS . ˆ (so U = U ˆ /π U ˆ ) and the ordinary character of U ˆ is (6) If U is liftable to an RG-lattice U χUˆ , then φU (g) = χUˆ (g) on p-regular elements g ∈ G. Proof. (1) In its action on U the identity has dimk U eigenvalues all equal to 1. They all lift to 1 and the sum of the lifts is dimk U . (2) This follows because g and xgx−1 have the same eigenvalues. (3) The eigenvalues of g −1 on U are the inverses of the eigenvalues of g on U , as are the eigenvlues of g on U ∗ (since here g acts by the inverse transpose matrix). The lifting ˆ lifts λ then λ ˆ −1 of roots of unity is a group homomorphism, so the result follows since if λ lifts λ−1 . (4) If g is a p-regular element then U and V have bases u1 , . . . , ur and v1 , . . . , vs consisting of eigenvectors of g with eigenvalues λ1 , . . . , λr and µr , . . . , µs , respectively. Now the tensors ui ⊗ uj form a basis of eigenvectors of U ⊗ V with eivenvalues λi µj . Their lifts ˆ ˆj since lifting is a group homomorphism, and P λ ˆ ˆ j = (P λ ˆ P µ are λd i µ j = λi µ i,j i µ i i )( j ˆj ) so that φU ⊗V (g) = φU (g)φV (g). (5) If g is a p-regular element then khgi is semisimple so that V ∼ = U ⊕ W as khgimodules. It follows that the eigenvalues of g on V are the union of the eigenvalues on V and on W (taken with multiplicity), and from this φV (g) = φU (g) + φW (g) follows. If U ∼ = U1 we may consider the sequence 0 → U1 → U → 0 → 0 to see that φU = φU1 . The final sentence follows by an inductive argument. ˆ then g acts on U = U ˆ /π U ˆ (6) If g is p-regular and acts with eigenvalues µ1 , . . . , µn on U with eigenvalues µ1 + (π), . . . , µn + (π). Since φU (g) is the sum of the lifts of these last quantities we have φ(g) = µ1 + · · · µn = χUˆ (g). We may use these elementary properties to construct tables of Brauer characters. There are two significant tables which we might construct: the table of values of Brauer characters of simple modules, and the table of values of Brauer characters of indecomposable projective modules. By Theorem ? both these tables are square. We will eventually establish that they satisfy orthogonality relations which generalize those for ordinary characters, but we first present some examples. Examples. 1. Let G = S3 . We have seen that in both characteristic 2 and characteristic 3 the simple representations of S3 lift to characteristic zero, and so the Brauer characters of the simple modules form tables which are part of the ordinary character table of S3 . The indecomposable projective modules for a group always lift to characteristic zero, but if we do not have some information such as the Cartan matrix or the decomposition matrix it is hard to know a priori what their characters might be. In the case of S 3 we have already computed this information, and we now present the tables of Brauer characters of the simple and indecomposable projective modules.

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() (12) (123) 1 1 1 1 −1 1 2 0 −1

() (123) 1 1 2 −1

126

() (123) 2 2 2 −1

TABLE: The character tables of S3 , ordinary and in characteristic 2. () (12) 3 1 3 −1

() (12) 1 1 1 −1

TABLE: The character tables of S3 in characteristic 3. 2. When G = S4 the ordinary character table is 24 4 8 4 3 1 (12) (12)(34) (1234) (123) 1 1 1 1 1 1 −1 1 −1 1 2 0 2 0 −1 3 −1 −1 1 0 3 1 −1 −1 0 as seen in Example 3.11. In characterstic 2, S4 has two simple modules, namely the trivial module and the 2-dimensional module of S3 , made into a module for S4 via the quotient homomorphism S4 → S3 . Both of these lift to characteristic zero, and so the Brauer characters of the 2-modular representations are () (123) 1 1 2 −1 which is the same as for S3 . The Brauer characters of the reductions modulo 2 of the ordinary characters of S4 are () (123) 1 1 1 1 2 −1 3 0 3 0 and we see from this that the sign representation reduces to the trivial representation, and reductions of the two 3-dimensional representations each have the 2-dimensional representation and the trivial representation as composition factors with multiplicity 1. This is because there is a unique way to express their Brauer characters as linear combinations of

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the simple Brauer characters, and this determines the composition factors. It follows that the decomposition and Cartan matrices for S4 at the prime 2 are 

1 1  D = 0  1 1

 0 0  1  1 1

and

C=



1 1 0 0 0 1



1 1 1 1  0 1 1  1 1

 0 0  4  1 = 2  1 1

2 3



.

In characteristic 3 the trivial representation and the sign representation are distinct 1-dimensional representations, and we also have two non-isomorphic 3-dimensional representations which are the reductions modulo 3 of the two 3-dimensional ordinary representations. This is because these 3-dimensional representations are blocks of defect zero by ?, and they remain simple on reduction modulo 3. This constructs four simple representations in characteristic 3, and this is the complete list by ? because S4 has four 3-regular conjugacy classes. Thus the table of Brauer characters of simple modules in characteristic 3 is 1 (12) (12)(34) (1234) 1 1 1 1 1 −1 1 −1 3 −1 −1 1 3 1 −1 −1 and each is the reduction of a simple module from characteristic zero. The remaining 2dimensional ordinary representation has Brauer character values 2, 0, 2, 0 and since this Brauer character is uniquely expressible as a linear combination of simple characters, namely the trivial Brauer character plus the sign Brauer character, these two 1-dimensional modules are the composition factors of any reduction modulo 3 of the 2-dimensional representation. We see that the decomposition and Cartan matrices for S4 in characteristic 3 are   1 0 0 0   2 1 0 0 0 1 0 0   1 2 0 0 D =  1 1 0 0  and C = D T D =  . 0 0 1 0   0 0 1 0 0 0 0 1 0 0 0 1

In these examples we have exploited the fact that the Brauer characters of the simple representations are independent to obtain the composition factors of the reductions of other modules, since there was a unique linear combination of the simple Brauer characters equal to the reduction of each ordinary simple character. As far as the examples were concerned it was possible to observe that the Brauer characters were independent after computing them, but it is reassuring to know that this is always the case. We prove this now, deducing it as a consequence of orthogonality relations for Brauer characters.

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(10.2) PROPOSITION. Let (F, R, k) be a p-modular system in which R is complete, and let G be a finite groups. Suppose that P and U are finite-dimensional kG-modules and that P is projective. Then dim HomkG (P, U ) =

1 |G|

X

φP (g −1 )φU (g).

p−regular g∈G

Proof. We make use of the isomorphism HomkG (U, V ) ∼ = HomkG (U ⊗k V ∗ , k) whenever U and V are finite-dimensional kG-modules, which holds since boths sides are isomorphic to (U ∗ ⊗k V )G , using results 3.3 and 3.4. Now HomkG (P, U ) ∼ = HomkG (P ⊗ U ∗ , k) and ∗ d P ⊗ U ∗ is a projective kG-module by 8.4. Thus it lifts to a projective RG-lattice P ⊗ k U and we have dim HomkG (P, U ) = dim HomkG (P ⊗ U ∗ , k) ∗ d = rank HomRG (P ⊗ k U , R)

∗ d = dim HomF G (F ⊗R P ⊗ k U ,F) 1 X χF ⊗ P ⊗ (g −1 )χk (g). = ∗ R d kU |G| g∈G

=

1 X χF ⊗ P ⊗ (g −1 ). ∗ R d kU |G| g∈G

We claim that χF ⊗

R

dU ∗ (g P⊗

−1

k

)=



φP (g −1 )φU (g) if g is p-regular, 0 otherwise,

and from this the result follows. If g is not p-regular it has order divisible by p and the ∗ d character value is zero by Proposition 9.27, since P ⊗ k U is projective. When g is p-regular we calculate the character value by using the fact that it depends only on the structure of P and U as khgi-modules. Since khgi is a semisimple algebra both P and U ∗ are now projective, and they lift to Rhgi-lattices whose tensor product Pˆ ⊗R Uˆ∗ is isomorphic to ∗ d P⊗ k U as Rhgi-modules, since both of these are projective covers as Rhgi-modules of P ⊗k U ∗ . From this we see that χF ⊗

as required.

d ∗ (g

R P ⊗k U

−1

) = χ(F ⊗Pˆ )⊗(F ⊗Uˆ∗ ) (g −1 ) = χF ⊗Pˆ (g −1 )χF ⊗Uˆ∗ (g −1 ) = φP (g −1 )φU (g)

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There is a similarity between the last result and Corollary ? in the last section, which related the dimensions of homomorphisms between modules in characteristic 0 and in characteristic p. The subtlety here is that the kG-module U might not be liftable to characterstic 0, whereas in ? this was an assumption. It is convenient to interpret the formula of the last proposition in terms of an inner product on a space of functions, in a similar way to what we did with ordinary characters. Let p−reg(G) denote the set of conjugacy classes of p-regular elements of G, so that p−reg(G) ⊆ cc(G) where the latter denotes the set of all conjugacy classes of G. We may regard Brauer characters as elements of the vector space Cp−reg(G) of functions p−reg(G) → C. We define a Hermitian bilinear form on this vector space by hφ, ψi =

1 |G|

X

φ(g)ψ(g)

p−regular g∈G

and just as with the similarly-defined bilinear form on Ccc(G) we note that hφθ, ψi = hφ, θ ∗ ψi where θ ∗ (g) = θ(g), the complex conjugate. With the notation of this bilinear form the last result now states that if P and U are finite-dimensional kG-modules with P projective then dim HomkG (P, U ) = hφP , φU i. (10.3) THEOREM (Row orthogonality relations). Let G be a finite group and k a splitting field for G of characteristic p. Let S1 , . . . , Sn be a complete list of non-isomorphic simple kG-modules, with projective covers PS1 , . . . , PSn . Then the Brauer characters φS1 , . . . , φSn of the simple modules form a basis for Cp−reg(G) , as do also the Brauer characters φPS1 , . . . , φPSn of the indecomposable projective modules. These two bases are dual to each other with respect to the bilinear form, in that hφPSi , φSj i = δi,j . The bilinear form on Cp−reg(G) is non-degenerate. Proof. Everything follows from the formula hφPSi , φSj i = δi,j and the fact that the number of non-isomorphic simple modules equals the number of p-regular conjugacy classes Pn Pn of G. Thus if i=1 λj φSj = 0 we have 0 = hφPSi , i=1 λj φSj i = λj , which shows that the φSj are independent, and hence form a basis. By a similar argument the φPSi also form a basis. The matrix of the bilinear form with respect to these bases is the identity matrix and it is non-degenerate.

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This result implies, of course, that the Brauer characters of the simple kG-modules are linearly independent as functions on the set of p-regular conjugacy classes of G, a fact we observed and used in the earlier examples. It means that they are sufficient to distinguish the composition factors of a module. (10.4) COROLLARY. Let U and V be finite-dimensional kG-modules, where k is a field of characteristic p. Then U and V have the same composition factors if and only if their Brauer characters φU and φV are equal. As an application we may deduce, for example, that if S is a simple kG-module then S∼ = S ∗ if and only if φS takes real values, since this is the condition that φS = φS = φS ∗ . It is also true from the theorem that the Brauer characters of the indecomposable projective modules are independent functions on p−reg(G), and so Brauer characters enable us to distinguish between projective modules. Since the Brauer characters of a module are determined by its composition factors we have the following consequence. (10.5) COROLLARY. Let P and Q be projective kG-modules, where k is a splitting field of characteristic p. Then P and Q are isomorphic if and only if they have the same composition factors. The Cartan matrix is invertible. The decomposition matrix has maximum rank. Pn Proof. Write P = PSa11 ⊕ · · · ⊕ PSann and Q = PSb11 ⊕ · · · ⊕ PSbnn so that φP = i=1 ai φPSi Pn and φQ = b φ . Then P ∼ = Q if and only if ai = bi for all i, if and only if i=1 Pn Pni PSi i=1 ai φPSi = i=1 bi φPSi since the φPSi are linearly independent, if and only if φP = φQ . By the last corollary this happens if and only if P and Q have the same composition factors. We claim that the kernel of the Cartan homomorphism c : K0 (kG) → G0 (kG) is zero. Any element of K0 (kG) can be written [P ] − [Q] where P and Q are projective modules, and such an element lies in the kernel if and only if P and Q have the same composition factors. This forces P ∼ = Q, so that the kernel is zero. It now follows that the Cartan homomorphism is an isomorphism. For the final statement about the decomposition matrix we use the fact that C = D T D. Thus rank C ≤ rank D. The number of columns of D equals the number of columns of C, and this number must be the rank of D. We may write the row orthogonality relations in various ways. The most rudimentary way is the statement that if S and T are simple kG-modules then 1 |G|

X

p−regular g∈G

φPT (g

−1

)φS (g) =

n

1 0

if S ∼ = T, otherwise.

We can also express this as a matrix product. Let Φ be the table of Brauer character values of simple kG-modules, Π the table of Brauer character values of indecomposable

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projective modules, and let B be the diagonal matrix whose entries are through the p-regular classes. The row orthogonality relations are

131 1 |CG (g)|

as g ranges

ΠBΦT = I. From this we obtain the column orthogonality relations. (10.6) PROPOSITION (Column orthogonality relations). With the above notation,   |C (x )| 0 ··· 0 G 1 0 |CG (x2 )|   T  Φ Π= . . .   .. .. .. 0 · · · |CG (xn )| where x1 , . . . , xn are representatives of the p-regular conjugacy classes of elements of G. Thus n X φS (g −1 )φPS (h) = |CG (g)| if g and h are conjugate, 0 otherwise. simple S The orthogonality relations can be used to determine the composition factors of a representation in a similar way to the procedure with ordinary characters. However this possibility is less useful than in characteristic zero because we need to know the Brauer characters of the indecomposable projectives to make it work. Usually we would only have this information once we have fairly complete information about the simple modules, whereas in characteristic zero we can test for simplicity of a character and subtract known character summands without complete character table information. The orthogonality relations do allow us to deduce important theoretical information about the Cartan and decomposition matrices. (10.7) COROLLARY. Let k be a splitting field of characteristic p for the finite group G. Let Φ be the table of values of Brauer characters of the simple kG-modules, Π the table of values of Brauer characters of the indecomposable projective kG-modules, C the Cartan matrix of G and D the decomposition matrix of G. Then (1) Π = CΦ, and (2) Φ, Π and C are invertible matrices. (3) D has maximum rank. A finer result than this is true. The determinant of the Cartan matrix over a splitting field of characteristic p is known to be a power of p, and the decomposition map d : G0 (F G) → G0 (kG) is a surjective homomorphism of abelian groups. Knowing that its matrix has maximum rank implies only that the cokernel is finite. These results may be proved by a line of argument which originates with the induction theorem of Brauer, a

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result which we have skipped over. The surjectivity of d is in turn implied by the FongSwan theorem in the case of p-solvable groups. We may interpret the cde triangle in terms of Brauer characters. As before, we let cc(G) denote the set of conjugacy classes of G and p−reg(G) the set of p-regular conjugacy classes of G. There are group homomorphisms K0 (kG) → Cp−reg(G)

G0 (kG) → Cp−reg(G)

G0 (F G) → Ccc(G)

defined on the basis elements [P ] ∈ K0 (kG), [S] ∈ G0 (kG) and [T ] ∈ G0 (F G), by [P ] 7→ φP ,

[S] 7→ φS ,

and

[T ] 7→ χT .

In fact these same formulas hold whenever P is an arbitrary finitely-generated pronective module and S, T are arbitrary finitely-generated modules. We may regard each of K0 (kG), G0 (kG) and G0 (F G) as lattices in complex vector spaces obtained by extending the scalars from Z to C. Because each of the above assignments is a correspondence of basis elements, the resulting linear maps they define are isomorphisms of vector spaces C ⊗Z K0 (kG) ∼ = Cp−reg(G) C ⊗Z G0 (kG) ∼ = Cp−reg(G)

C ⊗Z G0 (F G) ∼ = Ccc(G) .

Thus the cde triangle, on extending scalars to C, yields a triangle of vector spaces Ccc(G) e

d

% Cp−reg(G)

& c

−→

Cp−reg(G)

where, as before, the maps c, d, e have matrices C, D T and D, respectively. (10.8) PROPOSITION. Let (F, R, k) be a splitting p-modular system for G. With this interpretation of the cde triangle, if φ ∈ Cp−reg(G) then e(φ) is the function which is the same as φ on p-regular conjugacy classes and is zero on the other conjugacy classes. If χ ∈ Ccc(G) then d(χ) is the restriction of χ to the p-regular conjugacy classes. The Cartan homomorphism c is an isomorphism, the decomposition map d is surjective, and e is injective. The image of e is the space of class functions which are non-zero only on p-regular classes. Proof. The description of e(φ) follows from Proposition 9.27 and the definition of e, whereas the description of d comes from part (6) of Proposition 10.1. The Cartan

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homomorphism is an isomorphism because the Cartan matrix is non-singular, and because the decomposition matrix D has maximal rank d is surjective and e is injective. The image of e is a space whose dimension is the number of p-regular conjugacy classes of G, and it is contained in the space of maps whose support is the set of p-regular conjugacy classes, so we must have equality. Exercises for Section 10. 1. If θ ∈ Cp−reg(H) is a function defined on the set of p-regular conjugacy classes we may define an induced function θ ↑G H by means of the same formula in Proposition 4.10 which was used to define induction of ordinary class functions. Let H be a subgroup of G and let U be a finite-dimensional kH-module with Brauer character φU , where k is a G G field of characteristic p. Prove that φU ↑G = φU ↑G H , and that e(θ ↑H ) = e(θ) ↑H and H G d(χ ↑G H ) = d(χ) ↑H if χ is a class function. p−reg(G) Similarly define the restriction ψ ↓G and show that similar formulas H where ψ ∈ C hold. G Show that hθ ↑G H , ψi = hθ, ψ ↓H i always holds. 2. Let (F, R, k) be a splitting p-modular system for G. Let P and Q be finitelygenerated projective RG-modules such that F ⊗R P ∼ = F ⊗R Q as F G-modules. Show that ∼ P = Q. 3. The simple group GL(3, 2) has order 168 = 8 · 3 · 7. The following is part of its ordinary character table (the first two rows list the orders of centralizers of elements, and then underneath the orders of elements in the conjugacy classes): 168 1 1 3

8 2 1 -1

4 4 1 1

6 7 8

2 -1

0 -1

3 3 1

1 -1

7 7A 1 α

7 7B 1 α

-1

-1

1

1

Here α = ηˆ + ηˆ2 + ηˆ4 where ηˆ = e2πi/7 . Note that α2 = α ¯ − 1 and αα ¯ = 2. (a) Obtain the complete character table of GL(3, 2). (b) Compute the table of Brauer characters of simple F2 [GL(3, 2)]-modules. (c) Find the decomposition matrix and Cartan matrix of GL(3, 2) at the prime 2. (d) Write down the table of Brauer characters of projective F2 [GL(3, 2)]-modules. (e) Determine the direct sum decomposition of the module 8⊗3 (where 8 and 3 denote F2 [GL(3, 2)]-modules of those dimensions), as a direct sum of indecomposable modules. (f) Determine the composition factors of 3 ⊗ 3 and 3 ⊗ 3∗ , where 3 denotes the natural 3-dimensional F2 [GL(3, 2)]-module. [On approach is to use the orthogonality relations.]

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4. Let p be an odd prime. The center of SL(2, p) consists of the two scalar matrices {±I} (do not prove this), and the group P SL(2, p) is defined to be SL(2, p)/{±I}. The simple Fp [SL(2, p)]-modules thus consist of the simple Fp [P SL(2, p)]-modules, together with those simple modules on which −I acts non-trivially. The simple Fp [SL(2, p)]-modules were constructed in Exercise 21 of Section 6 as the symmetric powers S r (U2 ) of the natural 2-dimensional representation U2 where 0 ≤ r ≤ p − 1. Show that −I acts trivially on such a simple Fp [SL(2, p)]-module if and only if the module has odd dimension. Deduce that the simple Fp [P SL(2, p)]-modules have dimensions 1, 3, 5, . . . , p, constructed as the even symmetric powers of the 2-dimensional Fp [SL(2, p)]-module. 5. It so happens that GL(3, 2) ∼ = P SL(2, 7). (a) Construct the table of Brauer characters of simple F7 [P SL(2, 7)]-modules. [It will help to observe that an element of order 8 in SL(2, 7) represents an element of order 4 in P SL(2, 7), and its square represents an element of order 2.] (b) Compute the decomposition and Cartan matrices for P SL(2, 7) in characteristic 7. Show that the projective cover of the trivial module, P1 , has just four submodules, namely 0, P1 and two others.

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11. Indecomposable modules Until now the indecomposable modules we have considered have mostly been the simple and projective modules, and for these we have obtained straightforward descriptions in the case of an algebra which is finite-dimensional over a field, or of finite rank over a complete discrete valuation ring. We now consider indecomposable modules in general and we start with some of their basic properties. We first present an extension of Proposition 7.4, combined with Lemma 2.2. (11.1) PROPOSITION. Let U be a module for a ring A with a 1. Expressions U = U1 ⊕ · · · ⊕ Un as a direct sum of submodules biject with expressions 1U = e1 + · · · + en for the identity 1U ∈ EndA (U ) as a sum of orthogonal idempotents. Here ei is the composite of projection and inclusion U → Ui → U , and Ui = ei (U ). The summand Ui is indecomposable if and only if ei is primitive. Proof. We must check several things. Two constructions are indicated in the statement of the proposition, whereby given a direct sum decomposition of U we obtain an idempotent decomposition of 1U , and vice-versa. It is clear that the idempotents constructed from a module decomposition are orthogonal and sum to 1U . Conversely, given an expression 1U = e1 + · · · + en as a sum of orthogonal idempotents, every element u ∈ U can be written u = e1 u + · · · + en u where ei u ∈ ei U = Ui . In any expression u = u1 + · · · un with ui ∈ ei U we have ej ui ∈ ej ei U = 0 if i 6= j so ei u = ei ui = ui , and this expression is uniquely determined. Thus the expression 1U = e1 + · · · + en gives rise to a direct sum decomposition. We see that Ui decomposes as Ui = V ⊕ W if and only if ei = eV + eW can be written as a sum of orthogonal idempotents, and so Ui is indecomposable if and only if ei is primitive. (11.2) COROLLARY. An A-module U is indecomposable if and only if the only non-zero idempotent in EndA (U ) is 1U . Proof. From the proposition, U is indecomposable if and only if 1U is primitive, and this happens if and only if 1U and 0 are the only idempotents in EndA (U ). This last implication in the forward direction follows since any idempotent e gives rise to an expression 1U = e + (1U − e) as a sum of orthogonal idempotents.

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Our next step is to pin down the structure of the endomorphism ring of an indecomposable module.

(1) (2) (3) (4) (5) (6)

(11.3) PROPOSITION. Let B be a ring with 1. The following are equivalent. B has a unique maximal left ideal. B has a unique maximal right ideal. B/ Rad(B) is a division ring. The set of elements in B which are not invertible forms a left ideal. The set of elements in B which are not invertible forms a right ideal. The set of elements in B which are not invertible forms a 2-sided ideal.

Proof. (1) ⇒ (3) Let I be the unique maximal left ideal of B. Since Rad(B) is the intersection of the maximal left ideals, it follows that I = Rad(B). If a ∈ B − I then Ba is a left ideal not contained in I, so Ba = B. Thus there exists x ∈ B with xa = 1. Furthermore x 6∈ I, so Bx = B also and there exists y ∈ B with yx = 1. Now yxa = a = y so a and x are 2-sided inverses of one another. This implies that B/I is a division ring. (1) ⇒ (6) The argument just presented shows that the unique maximal left ideal I is in fact a 2-sided ideal, and every element not in I is invertible. This implies that every non-invertible element is contained in I. Equally, no element of I can be invertible, so I consists of the non-invertible elements, and they form a 2-sided ideal. (3) ⇒ (1) If I is a maximal left ideal of B then I ⊇ Rad(B) and so corresponds to a left ideal of B/ Rad(B), which is a division ring. It follows that either I = Rad(B) or I = B, and so Rad(B) is the unique maximal left ideal of B. (4) ⇒ (1) Let J be the set of non-invertible elements of B and I a maximal left ideal. Then no element of I is invertible, so I ⊆ J . Since J is an ideal, we have equality, and I is unique. (6) ⇒ (4) This implication is immediate, and so we have established the equivalence of conditions (1), (3), (4) and (6). Since conditions (3) and (6) are left-right symmetric, it follows that they are also equivalent to conditions (2) and (5), by analogy with the equivalence with (1) and (4). We will call a ring B satisfying any of the equivalent conditions of the last proposition a local ring. Any commutative ring which is local in the usual sense (i.e. it has a unique maximal ideal) is evidently local in the non-commutative sense. We see also that if G is a p-group and k is a field of characteristic p then the group algebra kG is a local ring, because its radical is the augmentation ideal and the quotient by the radical is k, which is a division ring. In our application of 11.3 to the proof of the Krull-Schmidt theorem we will want to know that in a local ring every element outside the radical is invertible. We point out also that if B is a ring with a 2-sided ideal I which is nilpotent and for which B/I is a division ring, then necessarily B is local. This will, at one point, be a useful way of showing that a ring is local. We also see that in a local ring B, the only idempotents are 0 and 1. (Any

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other idempotent would give a direct sum decomposition of B, and hence more than one maximal left ideal.) Combining this with Corollary 11.2, any module which has a local endomorphism ring must necessarily be indecomposable, without restriction on the ring for which this is a module. In special circumstances we obtain also the converse implication, as will now be shown. (11.4) PROPOSITION. Suppose that B is an R-algebra which is finitely generated as an R-module, where R is a complete discrete valuation ring or a field. Then B is a local ring if and only if the only non-zero idempotent in B is 1. Proof. We have just observed that if B is a local ring then the only idempotents are 0 and 1. Conversely, suppose that 0 and 1 are the only idempotents in B, and let (π) be the maximal ideal of R. Just as in the proof of part (1) of Proposition 9.12 we see that π annihilates every simple B-module, and so πB ⊆ Rad(B). This implies that B/ Rad(B) is a finite-dimensional R/(π)-algebra. If e ∈ B/ Rad(B) is idempotent then by the argument of Proposition 9.14 it lifts to an idempotent of B, which must be 0 or 1. Since e is the image of this lifting, it must also be 0 or 1. Now B/ Rad(B) ∼ = Mn1 (∆1 ) ⊕ · · · ⊕ Mnt (∆t ) for certain division rings ∆i , since this is a semisimple algebra, and the only way this algebra would have just one non-zero idempotent is if t = 1 and n1 = 1. This shows that condition (3) of the last proposition is satisfied. (11.5) COROLLARY. Let U be a finitely-generated RG-module where R is a complete discrete valuation ring or a field and G is a finite group. Then U is indecomposable if and only if EndRG (U ) is a local ring. Proof. Putting together the last results, all we need to do is to show that EndRG (U ) is finitely-generated as an R-module. Let Rm → U be a surjection of R-modules. Composition with this surjection gives a homomorphism EndRG (U ) → HomR (Rm , U ), and it is an injection since Rm → U is surjective. Thus EndRG (U ) is realized as an R-submodule of HomR (Rm , U ) ∼ = U m , which is a finitely generated R-module. Since R is Noetherian, the submodule is also finitely-generated. (11.6) THEOREM (Krull-Schmidt). Let A be a ring with a 1, and suppose that U is an A-module which has two A-module decompositions U = U 1 ⊕ · · · ⊕ U r = V1 ⊕ · · · ⊕ V s where for each i, EndA (Ui ) and EndA (Vi ) are local rings. Then r = s and the summands Ui and Vj are isomorphic in pairs when taken in a suitable order. Proof. The proof is by induction on max{r, s}. When this number is 1 we have U = U1 = V1 , and this starts the induction.

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Now suppose max{r, s} > 1 and the result is true for smaller values of max{r, s}. For each j let πj : U → Vj be projection onto the jth summand with respect to the Ps decomposition U = V1 ⊕ · · · ⊕ Vs , and let ιj : Vj ,→ U be inclusion. Then j=1 ιj πj = 1U . Now let β : U → U1 be projection with respect to the decomposition U = U1 ⊕ · · · ⊕ Ur and α : U1 ,→ U be inclusion so that βα = 1U1 . We have 1U1 = β(

s X j=1

ιj πj )α =

s X

βιj πj α

j=1

and since EndA (U1 ) is a local ring it follows that at least one term βιj πj α must be invertible. By renumbering the Vj if necessary we may suppose that j = 1, and we write φ = βι1 π1 α. Now (φ−1 βι1 )(π1 α) = 1U1 and so π1 α : U1 → V1 is split mono and φ−1 βι1 : V1 → U1 is split epi. It follows that π1 α(U1 ) is a direct summand of V1 . Since EndA (V1 ) is local, V1 is indecomposable and so π1 α(U1 ) = V1 . Thus π1 α : U1 → V1 is an isomorphism. We now show that U = U1 ⊕ V2 ⊕ · · · ⊕ Vs . Because π1 α is an isomorphism, π1 is one-to-one on the elements of U1 . Also π1 is zero on V2 ⊕ · · · ⊕ Vs and it follows that U1 ∩ (V2 ⊕ · · · ⊕ Vs ) = 0, since any element of the intersection is detected by its image under π1 , and this must be zero. The submodule U1 + V2 + · · · + Vs contains V2 + · · · + Vs = Ker π1 and so corresponds via the first isomorphism theorem for modules to a submodule of π1 (U ) = V1 . In fact π1 is surjective and so U1 + V2 + · · · + Vs = U . It follows that U = U1 ⊕ V2 ⊕ · · · ⊕ Vs . We now deduce that U/U1 ∼ = U2 ⊕ · · · ⊕ U r ∼ = V2 ⊕ · · · ⊕ Vs . It follows by induction that r = s and the summands are isomorphic in pairs, which completes the proof. (11.7) COROLLARY. Let R be a complete discrete valuation ring or a field and G a finite group. Suppose that U is a finitely-generated RG-module which has two decompositions U = U 1 ⊕ · · · ⊕ U r = V1 ⊕ · · · ⊕ V s where the Ui and Vj are indecomposable RG-modules. Then r = s and the summands Ui and Vj are isomorphic in pairs when taken in a suitable order.

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Relative projectivity Let H be a subgroup of G and R a commutative ring with 1. An RG-module is said to be H-free if it has the form V ↑G H for some RH-module V . It is H-projective, or projective relative to H, if it is a direct summand of a module of the form V ↑G H for some RH-module V. For example, the regular representation RG ∼ = R ↑G 1 is 1-free, and projective modules are 1-projective. If R is a field then every 1-projective module is projective, but if R is not a field and V is an R-module which is not projective as an R-module, then V ↑G 1 is 1-free but not projective as an RG-module. Every RG-module is G-projective. In order to investigate relative projectivity we first deal with some technicalities. We P have seen the pervasive importance of the group ring element g∈G g at every stage of the development of representation theory. As an operator on any representation of G it has image contained in the G-fixed points. We now consider something more general, and for H a subgroup H of G and an RG-module U we define the relative trace map trG → U G. H :U To define this we choose a set of representatives g1 , . . . , gt of the left cosets of H in G, so Pt G = g1 H ∪ · · · ∪ gt H. If u ∈ U H we define trG H (u) = i=1 gi u. (11.8) LEMMA. The homomorphism trG H is well-defined, and if K ≤ H ≤ G are G H G subgroups then trH trK = trK .

(11.9) LEMMA. Suppose that α : U → V and γ : W → X are homomorphisms G of RG-modules and that β : V ↓G H → W ↓H is an RH-module homomorphism. Then G G G (trG H β) ◦ α = trH (β ◦ α) and γ ◦ (trH β) = trH (γ ◦ β). Proof. Let g1 , . . . , gn be a set of left coset representatives for H in G and u ∈ U . Pn Pn −1 −1 G Then (trG = αu) = H β)α(u) i=1 gi β(gi P i=1 gi βα(gi u) = trH (βα)(u). Similarly P n n −1 −1 G γ(trG H β)(u) = γ i=1 gi β(gi u) = i=1 gi γβ(gi u) = trH (γβ)(u). (11.10) COROLLARY. The image of trG H : EndRH (U ) → EndRG (U ) is an ideal.

It will help us to consider the adjoint properties of induction and restriction of modules in detail. We have seen in result 4.12 that when U is an RH-module and V is an RGG ∼ module, where H ≤ G, we have HomRG (U ↑G H , V ) = HomRH (U, V ↓H ). There may be many such isomorphisms, but there is a choice which is natural in U and V . This means that whenever U1 → U2 is an RG-module homomorphism the resulting square HomRG (U1 ↑G H , V ) −→ x  

HomRG (U2 ↑G H , V ) −→

HomRH (U1 , V ↓G H) x  

HomRH (U2 , V ↓G H)

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commutes, as does the square G HomRG (U ↑G H , V1 ) −→ HomRH (U, V1 ↓H )     y y

G HomRG (U ↑G H , V2 ) −→ HomRH (U, V2 ↓H )

whenever V1 → V2 is a homomorhism of RG-modules. In this situation we say that G the operation ↑G H : RH-modules → RG-modules is left adjoint to ↓H : RG-modules → RH-modules. (These ‘operations’ are in fact functors.) We say also that ↓G H is right adjoint to ↑G . H We now describe explicitly a natural isomorphism between these groups of homomorphisms. For convenience we take a set of left coset representatives of H in G with g 1 = 1. Ln G For each RH-module U let µ : U → U ↑G H ↓H = i=1 gi ⊗ U be the inclusion into the G G summand 1 ⊗ U , so µ(u) = 1 ⊗ u, and let ν : U ↑H ↓H → U be projection onto this sumG mand. If V is an RG-module we define RG-module homomorphisms η : V → V ↓G H ↑H and P P P n −1 G  : V ↓G λx x ⊗ u) = λx xu. In fact, regarding H ↑H → V by η(v) = i=1 gi ⊗ gi v and ( G G G G G trH : HomRH (V, V ↓H ↑H ) → HomRG (V, V ↓H ↑H ) we have η = trG H µ where µ has domain V regarded as an RH-module. Similarly  = trG ν, and this shows that η and  are defined H independently of the choice of coset representatives. We see that η is a monomorphism,  is an epimorphism and their composite is multiplication by |G : H|. Given an RG-module homomorphism U ↑G H → V we obtain an RG-module homoµ α G G G morphsim U −→U ↑H ↓H −→V ↓H , and given an RH-module homomorphism β : U → β↑G



H G G G V ↓G H we obtain an RG-module homomorphism U ↑H −→V ↓H ↑H −→V . We may check that these two constructions are mutually inverse, and are natural in U and V . Group rings of finite groups have the special property that not only is induction left adjoint to restriction, it is also right adjoint. This coincidence of the left and right adjoint of restriction is related to other phenomena we have studied, such as the fact that over a field projective and injective modules are the same thing. We construct a natural isomorphism

G ∼ HomRG (V, U ↑G H ) = HomRH (V ↓H , U )

whenever U is an RH-module and V is an RG-module. Given an RG-module homomorphism α : V → U ↑G H we obtain an RH-module homomorphism α

ν

G G V ↓G H −→U ↑H ↓H −→U

and given an RH-module homomorphism β : V ↓G H → U we obtain an RG-module homomorphism η

β↑G

H G G V −→V ↓G H ↑H −→U ↑H .

Again we check that these operations are natural and mutually inverse.

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(11.11) PROPOSITION. Let G be a finite group with a subgroup H. The following are equivalent for an RG-module U . (1) U is H-projective. (2) Whenever we have homomorphisms U  ψ y

φ

V

−→

W

where φ is an epimorphism and for which there exists an RH-module homomorphism G U ↓G H → V ↓H making the diagram commute, then there exists an RG-module homomorphism U → V making the diagram commute. G (3) Whenever φ : V → U is a homomorphism of RG-modules such that φ ↓G H : V ↓H → U ↓G H is a split epimorphism of RH-modules, then φ is a split epimorphism of RGmodules. (4) The surjective homomorphism of RG-modules G U ↓G H ↑H = RG ⊗RH U → U

x ⊗ u 7→ xu

is split. G (5) U is a direct summand of U ↓G H ↑H . (6) (Higman’s criterion) 1U lies in the image of trG H : EndRH (U ) → EndRG (U ). Proof. (1) ⇒ (2) We first prove this implication in the special case when U is an induced module T ↑G H . Suppose we have a diagram of RG-modules

V

φ

−→

T ↑G H  ψ y W

G G and a homomorphism of RH-modules α : T ↑G H ↓H → V ↓H so that ψ = φα. Under the µ

ψ

G G adjoint correspondence ψ corresponds to the composite T −→T ↑G H ↓H −→W ↓H and we have a commutative triangle of RH-module homomorphisms

αµ

.

V ↓G H

φ↓G

T  ψµ y .

H −→ W ↓G H

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By the adjoint correspondence (and its naturality) this corresponds to a commutative triangle of RG-module homomorphisms T ↑G H  ψ y

(αµ)↑G H

. φ

V

−→

W

which proves this implication for the module T ↑G H. ι G π Now consider a module U which is a summand of T ↑G H , and let U −→T ↑H −→U be G inclusion and projection. We suppose there is a homomorphism α : U ↓G H → V ↓H so that G φα = ψ. The homomorphism ψπ : T ↑H → W has the property that ψπ = φ(απ) and so by what we proved there is a homomorphism of RG-modules β : T ↑G H → V so that φβ = ψπ. Now φβι = ψπι = ψ so that βι : U → V is an RG-module homomorphism which makes the triangle commute. (2) ⇒ (3) This follows immediately on applying (2) to the diagram

V

−→

U  1 y .U

U

G (3) ⇒ (4) We know that  : U ↓G H ↑H → U is split as an RH-module homomorphism G by µ : U → U ↓G H ↑H . Applying condition (3) it splits as an RG-module homomorphism. (4) ⇒ (5) and (5) ⇒ (1) are immediate. G (5) ⇒ (6) We may prove that 1U ↓G ↑G = trH (µν) by direct computation. Writing H H G U ↓G H ↑H = V1 ⊕ V2 where V1 = U , we can represent µν as a matrix   f11 f21 µν = f12 f22

where fij : Vi → Vj . Then trG H (µν)

=



trG H f11 trG H f12

trG H f21 trG H f22



=



1 0

0 1



G and from this we see that for every summand of U ↓G H ↑H (and in particular for U ) the G identity map on that summand is in the image of trH . G G (6) ⇒ (5) Write 1U = trG H α for some morphism α : U ↓H → U ↓H . Now α corresponds by the adjoint correspondence to the composite homomorphism η

α↑G

H G G G U −→U ↓G H ↑H −→U ↓H ↑H .

We claim that α ↑G H η splits : for

G G G G G α ↑G H η = trH (ν)α ↑H η = trH (να ↑H η) = trH (α) = 1U .

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Condition (6) is in fact equivalent to the statement that trG H : EndRH (U ) → EndRG (U ) G is surjective. This is because the image of trH is an ideal in EndRG (U ), and so it equals EndRG (U ) if and only if it contains 1U . Conditions (2), (3) and (4) are modeled on conditions associated with the notion of projectivity. There are dual conditions associated with the notion of an injective module, obtained by reversing the arrows and interchanging the words ‘epimorphism’ and ‘monomorphism’. These conditions are also equivalent to the ones stated in this result. In fact, the notion of relative projectivity in the context of group algebras of finite groups might have been termed relative injectivity with equal validity. (11.12) PROPOSITION. Suppose that H is a subgroup of G and that |G : H| is invertible in the ring R. Then every RG-module is H-projective. Proof. For any RG-module U we may write 1U = tion (6) of the last result.

1 |G:H|

trG H 1U , thus verifying condi-

(11.13) COROLLARY. Suppose that H is a subgroup of G for which |G : H| is invertible in the ring R, and let U be an RG-module. Then U is projective as an RGmodule if and only if U ↓G H is projective as an RG-module. Proof. We already know that if U is projective then U ↓G H is projective, no matter G what subgroup H is. conversely, if U ↓H is projective it is a summand of a free module G RH n . Since U is H-projective it is a summand of U ↓G H ↑H , which is a summand of n ∼ RH n ↑G H = RG . Therefore U is projective. This criterion for projectivity would have simplified matters when we were considering the projective modules for groups of the form G = H o K in Section 8. In this situation we saw that RG becomes an RG-module where H acts by left multiplication and K acts by conjugation. If K has order prime to p and R is a field of characteristic p (or a discrete valuation ring with residue field of characteristic p) it follows from the corollary that RH is projective as an RG-module, because on restriction to H it is projective and the index of H in G is invertible. On the other hand, we may also regard RK as an RG-module via the homomorphism G → K, and if now H has order prime to p then RK is a projective RG-module, because it is projective on restriction to RK and the index of K in G is prime to p. In the exercises to Section 6 the simple Fp SL2 (p)-modules were considered. The goal of Exercise 21 of Section 6 was to show that the symmetric powers S r (U2 ) are all simple Fp SL2 (p)-modules when 0 ≤ r ≤ p − 1, where U2 is the 2-dimensional space on which SL2 (p) acts as invertible transformations of determinant 1. The order of SL2 (p) is p(p2 −1) and so a Sylow p-subgroup of this group is cyclic of order p. In Exercise 20 of Section 6 one shows that on restriction to a certain Sylow p-subgroup, S r (U2 ) is indecomposable of dimension r + 1 when 0 ≤ r ≤ p − 1. From the classification of indecomposable modules for a cyclic group of order p we deduce that S p−1 (U2 ) is projective as a module for the

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Sylow p-subgroup. It follows from the last corollary that S p−1 (U2 ) is projective as an Fp SL2 (p)-module. This module is thus a simple projective Fp SL2 (p)-module, or in other words a block of defect zero.

Representation type of algebras In trying to understand the representation theory of a ring one hopes, where possible, to be able to describe all the indecomposable modules. The possibility of such a description depends on there being in some sense sufficiently few indecomposable modules for a classification to be a reasonable goal, or for a description of the indecomposable modules to make any sense or have any utility. Unfortunately for the majority of rings one encounters both a classification of the indecomposable modules and a way of organizing the classification so that it can be understood seem to be unreasonable expectations. On the other hand, in special cases the indecomposable modules can indeed be classified. With this in mind, we say that a ring A has finite representation type if and only if there are only finitely many isomorphism classes of indecomposable A-modules, and otherwise we say that R has infinite representation type. We have seen in Theorem 6.2 that if G is a cyclic p-group and k is a field of characteristic p, then kG has finite representation type. (11.14) PROPOSITION. Let R be a discrete valuation ring with residue field of characteristic p or a field of characteristic p, and let P be a Sylow p-subgroup of a finite group G. Then RG has finite representation type if and only if RP has finite representation type. Proof. Since |G : P | is invertible in R, by Proposition ? every indecomposable RGmodule is a summand of some module T ↑G P , and we may assume that T is indecomposable, G G since if T = T1 ⊕ T2 then T ↑P = T1 ↑P ⊕T2 ↑G P , and by the Krull-Schmidt theorem the G indecomposable summands of T ↑P are the indecomposable summands of T1 ↑G P together with the indecomposable summands of T2 ↑G P . If RP has finite representation type then G there are only finitely many modules T ↑P with T indecomosable, and these have only finitely many summands by the Krull-Schmidt theorem. Conversely, every RP -module U G G is a direct summad of U ↑G P ↓P and hence is a direct summand of some module V ↓P . If U is indecomposable, we may assume V is indecomposable. Now if RG has finite representation type there are only finitely many isomorphism types of summands of modules V ↓G P , by the Krull-Schmidt theorem, and hence RP has finite representation type.

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In preparation for a characterization of group algebras of finite representation type we now show that k[Cp × Cp ] has infinitely many non-isomorphic indecomposable modules, where k is a field of characteristic p. We will first describe infinitely many modules, and after that we will prove that they are indecomosable. Let G = Cp × Cp = hai × hbi and let k be a field of characteristic p. We define a module M2n+1 of dimension 2n + 1 with basis u1 , . . . , un , v0 , . . . , vn and an action of G given as follows: a(ui ) = ui + vi−1 , b(ui ) = ui + vi a(vi ) = vi , b(vi ) = vi

where 1 ≤ i ≤ n where 0 ≤ i ≤ n.

It is perhaps easier to write this as (a − 1)ui = vi−1 , (b − 1)ui = vi (a − 1)vi = 0, (b − 1)vi = 0

where 1 ≤ i ≤ n where 0 ≤ i ≤ n

and to describe M2n+1 diagrammatically u1 a−1

.

v0

u2 b−1

a−1

&

.

un b−1

&

···

a−1

.

b−1

&

v1

vn

We may check that this is indeed a representation of G by verifying that (a − 1)(b − 1)x = (b − 1)(a − 1)x = 0 and (a − 1)p x = (b − 1)p x = 0 for all x ∈ M2n+1 , which is immediate. Let us now show that M2n+1 is indecomposable. We will do this by showing that EndkG (M2n+1 )/I has dimension 1 for a certain nilpotent ideal I. Observe that Soc(M2n+1 ) = Rad(M2n+1 ) = kv0 + · · · kvn . The ideal I in question is HomkG (M2n+1 , Soc(M2n+1 )), and this squares to zero since if φ : M2n+1 → Soc(M2n+1 ) then Rad(M2n+1 ) ⊆ Ker φ and so φ Soc(M2n+1 ) = 0. If φ is any endomorphism of M2n+1 then φ(Soc(M2n+1 )) ⊆ Soc M2n+1 so φ induces an endomorphism φ¯ of M2n+1 / Soc(M2n+1 ). We show that φ¯ is necessarily a scalar multiple of the identity. To establish this we will exploint the equations (a − 1)ui = (b − 1)ui−1

when 2 ≤ i ≤ n

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and also the fact that (a − 1) and (b − 1) both map ku1 + · · · + kun injectively into Soc(M2n+1 ). We have φ((a − 1)ui ) = (a − 1)φ(ui ) = φ((b − 1)ui−1 ) = (b − 1)φ(ui−1 )

and it follows from this that φ¯ is completely determined once we know φ(u1 ), since then ¯ 2 ), (a − 1)φ(u3 ) = (b − 1)φ(u2 ) determines (a − 1)φ(u2 ) = (b − 1)φ(u1 ) determines φ(u ¯ 3 ), and so on. φ(u Suppose that φ(u1 ) ≡ λ1 u1 + · · · + λr ur (mod Soc(M2n+1 )) where the λi are scalars and λr 6= 0. The equations imply that φ(u2 ) ≡ λ1 u2 + · · · + λr ur+1 (mod Soc(M2n+1 )) and inductively φ(un−r+1 ) ≡ λ1 un−r+1 + · · · + λr un (mod Soc(M2n+1 )). If it were the case that r > 1 then the equation (b − 1)φ(un−r+1 ) = (a − 1)φ(un−r+2 ) = λ1 vn−r+1 + · · · + λr vn would have no solution, since no such vector where the coeffiecient of vn is non-zero lies in the image of a − 1. We conclude that r = 1 and φ(ui ) ≡ λ1 ui (mod Soc(M2n+1 )) for some scalar λ1 , for all i with 1 ≤ i ≤ n. Thus φ − λ1 1M2n+1 : M2n+1 → Soc(M2n+1 ) and so EndkG (M2n+1 )/ HomkG (M2n+1 , Soc(M2n+1 )) has dimension 1. We have proved (11.15) PROPOSITION. The quotient EndkG (M2n+1 )/ Rad EndkG (M2n+1 ) has dimension 1. Thus EndkG (M2n+1 ) is a local ring and M2n+1 is indecomposable. (11.16) THEOREM (D.G. Higman). Let k be a field of charactersitic p. Then kG has finite representation type if and only if Sylow p-subgroups of G are cyclic. Proof. By Proposition 11.14 it suffices to show that if P is a p-group then kP has finite representation type if and only if P is cyclic. We have seen in ? that kP has finite representation type when P is cyclic. If P is not cyclic then P has the group Cp × Cp as a homomorphic image. (This may be proved using the fact that if Φ(P ) is the Frattini subgroup of P then P/Φ(P ) ∼ = (Cp )d for some d and that P can be generated by d elements. Since P cannot be generated by a single element, d ≥ 2 and so (Cp )2 is an image of P .) The infinitely-many non-isomorphic indecomposable k[Cp × Cp ]-modules become non-isomorphic indecomposable kP -modules via the quotient homomorphism, and this establishes the result.

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Even when the representation type is infinite, the arguments that we have been using still yield the following result.

(11.17) THEOREM. Let k be a field of charactersitic p. For any finite group G, the number of isomorphism classes of indecomposable modules which are projective relative to a cyclic subgroup is finite.

We now consider the kinds of phenomena that can occur with indecomposable kGmodules. We will indicate for which groups we can expect a reasonable classification of indecomposable modules and point out some of the techniques which are available to describe them. An understanding of the indecomposable modules for a finite group in characteristic p is very much connected with an understanding of the modules for a Sylow p-subgroup. We generally expect there to be more modules for G than for its Sylow p-subgroup, and a description of modules for the Sylow p-subgroup tends to be easier than for G. We start with the easiest example of a description of indecomposable modules for a group algebra of infinite representation type, which is the modules for k[C2 × C2 ] where k is a field of characteristic 2. In constructing infinitely many indecomposable modules for Cp × Cp we already constructed some of the indecomposable k[C2 × C2 ]-modules, but now we complete the picture. As before we let G = C2 × C2 = hai × hbi and exhibit the modules diagrammatically by the action of a − 1 and b − 1 on a basis. Here are the indecomposable modules:

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Finite Group Representations

kG = •

. &

W2n+1 =

•

•

& .

•

&

W1 = M 1 = • M2n+1 = • Ef,n = • E0,n =

E∞,n =

• •

.

. . &

• •

•

•

148

•

.

•

&

& & .

• • •

•

&

•

.

. . &

• •

•

• •

···

···

··· ··· ···

• • •

•

.

•

&

. . &

•

•

• &

•

•

In these diagrams each node represents a basis element of a vector space, a southwest arrow . emanating from a node indicates that a − 1 sends that basis element to the basis element at its tip, and similarly a southeast arrow & indicates the action of b − 1 on a basis element. Where no arrow in some direction emanates from a node, the corresponding element a − 1 or b − 1 acts as zero. The even-dimensional indecomposable representations Ef,n require some further explanation. They are parametrized by pairs (f, n) where f ∈ k[X] is an irreducible monic polynomial and n ≥ 1 is an integer. Let the top row of nodes in the diagram correspond to basis elements u1 , . . . , un , and the bottom row to basis elements v1 , . . . , vn . Let (f (X))n = X mn + amn−1 X mn−1 + · · · + a0 . The right-most arrow & starting at umn which has no terminal node is supposed to indicate that (b − 1)umn = vmn+1 where vmn+1 = −amn−1 vmn−1 − · · · − a1 v2 − a0 v1 , so that with respect to the given bases b − 1 has matrix   0 1 ··· 0 .. .. ..  ...  . . .    0  0 1 −a0 · · · −amn−1

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which is an indecomposable matrix in rational canonical form with characteristic polynomial f n . (11.18) THEOREM. Let k be a field of characteristic 2. The k[C2 × C2 ]-modules shown are a complete list of indecomposable modules. Proof. We describe only the strategy of the proof, and refer to the exercises at the end of this section and [Ben] for more details. The first step is to use the fact that the regular P representation is injective, with simple socle spanned by g∈G g. If U is an indecomposable P P module for which ( g∈G g)U 6= 0 then there is a vector u ∈ U with ( g∈G g)u 6= 0. The homomorphism kG → U specified by x 7→ xu is a monomorphism, since if its kernel were P non-zero it would contain g∈G g, but this element does not lie in the kernel. Since kG is injective, the sumodule kGu is a direct summand of U , and hence U ∼ = kG since U is indecomposable. From this we deduce that apart from the regular representation, every P indecomposable module is annihilated by g∈G g, and hence is a module for the ring P kG/( g∈G g), which has dimension 3 and is isomorphic to k[α, β]/(α2 , αβ, β 2 ), where α corresponds to a − 1 ∈ kG and β to b − 1 ∈ kG. Representations of this ring are the same thing as the specification of a vector space U with a pair of linear endomorphisms α, β : U → U which annihilate each other and square to zero. The classification of such pairs of matrices up to simultaneous conjugacy of the matrices (which is the same as isomorphism of the module) was achieved by Kronecker in the 19th century, and he obtained the indecomposable forms which we have listed. The modules M2n+1 and W2n+1 have become known as string modules and the Ef,n as band modules, in view of the form taken by the diagrams which describe them. More complicated classifications, but with a similar flavor, have been achieved for representations of the dihedral, semidihedral and generalized quaternion 2-groups in characteristic 2. For dihedral 2-groups, all the modules apart from the regular representation are string modules or band modules (see ?). Provided the field k is infinite, k[C2 × C2 ] has infinitely many isomorphism types of indecomposable modules in each dimension larger than 1. They can nevertheless be grouped into finitely many families, as we have seen intuitively in their diagrammatic description. As a more precise version of this idea, consider the infinite-dimensional k[C2 × C2 ]-module M with diagram M= •

.

•

&

•

.

•

&

•

. ···

and basis u1 , u2 , . . . , u1 , v2 , . . . This module has an endomorphism η which shifts each of the two rows one place to the right, specified by η(ui ) = ui+1 and η(vi ) = vi+1 , so that M becomes a (k[C2 × C2 ], k[X])-bimodule, where the indeterminate X acts via η. As k[X]-modules we have M ∼ = k[X] ⊕ k[X]. Given an irreducible polynomial f ∈ k[X] and

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an integer n ≥ 1 we may construct the k[C2 × C2 ]-module M ⊗k[X] k[X]/(f n ), which is a module isomorphic to (k[X]/(f n))2 as a k[X]-module, and which is acted on by k[C2 × C2 ] as a module isomorphic to Ef, n. This construction accounts for all but finitely many of the indecomposable k[C2 × C2 ]-modules in each dimension. With our understanding improved by the last example, we now divide infinite representation type into two kinds: tame and wild. Let A be a finite-dimensional algebra over an infinite field k. We say A has tame representation type if it has infinite type and for each dimension d there are finitely many (A, k[X])-bimodules Mi which are free as k[X]-modules so that all but finitely many of the indecomposable A-modules of dimension d have the form Mi ⊗k[X] k[X]/(f n ) for some irreducible polynomial f and integer n. If the bimodules Mi can be chosen independently of d (as happens with representations of C2 × C2 ) we say that A has domestic representation type, and otherwise it is non-domestic. We say that the finite-dimensional algebra A has wild representation type if there is a finitely-generated (A, khX, Y i)-bimodule M which is free as a right khX, Y i)-module, such that the functor M ⊗khX,Y i from finite-dimensional khX, Y i-modules to finitedimensional A-modules preserves indecomposability and isomorphism type. Here khX, Y i is the free algebra on two non-commuting variables, having as basis the non-commutative monomials X a1 Y b1 X a2 Y b2 · · · where ai , bi ≥ 0. In view of the following theorem it would have been possible, over an algebraically closed field, to define wild to be everything which is not finite or tame. (11.19) THEOREM (Drozd [Dro]; Crawley-Boevey [CB]). Let A be a finite-dimensional algebra over an algebraically-closed field. Then A has either finite, tame or wild representation type. When A has wild representation type, the idea is that phenomena which occur with representations of khX, Y i also appear with A-modules. Thus A has at least as many isomorphism types indecomposable modules as khX, Y i does. The indecomposable khX, Y imodules are quite diverse, and in some sense it is a hopeless task to try to classify them. Thus, it is known that the theory of finite-dimensional khX, Y i-modules is undecidable, meaning that there exists a sentence in the language of finite-dimensional khX, Y i-modules which cannot be decided by any Turing machine (an account of this result can be found in [Prest]). It is also the case in a heuristic sense that the khX, Y i-modules are as badly behaved as those of any algebra: it is possible to embed the category of A-modules and their homomorphisms, for any algebra A, into the category of khX, Y i-modules (see [Brenner, Gabriel]). In the context of group algebras the division into finite, tame and wild representation type is given by the following theorem, in which we include the result of D.G. Higman already proven.

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(11.20) THEOREM (Bondarenko, Brenner, Drozd, Higman, Ringel). Let k be an infinite field of characteristic p and let G be a finite group with Sylow p-subgroup P . Then kG has finite representation type if and only if P is cyclic, tame representation type if and only if p = 2 and P is dihedral, semidihedral or generalized quaternion. In all other cases kG has wild representation type. We include C2 × C2 as a dihedral group in the statement of this theorem

The first step in the proof of this theorem we have already seen, and it is to identify the group algebras of finite representation type. Next, certain group algebras were established as being wild. This is implied ? by the following result, which also serves as an example of the kind of phenomenon we may expect with wild algebras. (11.21) THEOREM (Brenner [Bre1]). Let P be a finite p-group having either Cp ×Cp (p odd), C2 × C4 or C2 × C2 × C2 as a homomorphic image, let k be a field of characteristic p, and let E be a finite-dimensional algebra over k. Then there exists a finite-dimesnional kP -module M such that EndkP (M ) has a nilpotent ideal J and a subalgebra E 0 isomorphic to E, with the property that the quotient map sends E 0 isomorphically to EndkP (M )/J . A theorem of Blackburn implies that if P is a 2-group which is not cyclic and does not have C2 × C4 or C2 × C2 × C2 as a homomorphic image, then P is dihedral, semidihedral or generalized quaternion. Groups with these as Sylow 2-subgroups were the only groups whose representation type was in question at this point. It was decided by classifying explicitly the indecomposable modules in these cases, and it was done by Bondarenko, Drozd and Ringel. A later approach can be found in the work of Crawley-Boevey [CB].

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Vertices, sources and Green correspondence Having just given an impression of the difficult of classifying indecomposable modules, we now explain some positive techniques which are available for understanding them better. (11.22) THEOREM. Let R be a field or a complete discrete valuation ring, and let U be an indecomposable RG-module. (1) There is a unique conjugacy class of subgroups Q of G which are minimal subject to the property that U is Q-projective. (2) Let Q be a minimal subgroup of G such that U is Q-projective. There is an indecomposable RQ-module T which is unique up to conjugacy by elements of NG (Q) such that U is a summand of T ↑G Q. Proof. (1) We offer two proofs of this result, one employing module-theoretic techniques, and the other a ring-theoretic approach. Both proofs exploit similar ideas, in which the Mackey formula is a key ingredient. First proof: we start by supposing that U is both H-projective and K-projective where G G G H and K are subgroups of G. Then U is a summand of U ↓G H ↑H and also of U ↓K ↑K , so it is also a summand of M G G G H K G U ↓G ↑ ↓ ↑ = ( g((U ↓G H H K K H ) ↓K g ∩H )) ↑K∩ gH ) ↑K g∈[K\G/H]

=

M

G ( g(U ↓G K g ∩H )) ↑K∩ gH

g∈[K\G/H]

using transitivity of restriction and induction, and now U must be a summand of some module induced from one of the groups K ∩ gH. If both H and K happen to be minimal subject to the condition that U is projective relative to these groups, we deduce that 0 K ∩ gH = K, so K ⊆ gH. Similarly H ⊆ g K for some g 0 and so H and K are conjugate. Second proof: we start the same way and suppose that U is both H-projective and G K-projective. We may write 1U = trG H α = trK β for certain α ∈ EndRH (U ) and β ∈ EndRK (U ). Now G G G G G 1U = (trG H α)(trK β) = trK ((trH α)β) = trK (trH (αβ)) =

X

trG K∩ gH (cg αβ).

g∈[K\G/H]

Since U is indecomposable its endomorphism ring is local and so some term trG K∩ gH (cg αβ) must lie outside the unique maximal ideal of EndRG (U ) and must be an automorphism. This implies that trG K∩ gH : EndR[K∩ gH] (U ) → EndRG (U ) is surjective, since the image of G trK∩ gH is an ideal, and so U is K ∩ gH-projective. We now deduce as in the first proof that if K and H are minimal subgroups relative to which U is projective, then H and K are conjugate.

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(2) Let Q be a minimal subgroup relative to which U is projective. We know that U G G is a summand of U ↓G Q ↑Q and hence it is a summand of T ↑Q for some indecomposable 0 summand T of U ↓G Q . Suppose that T is another indecomposable module for which U is a L Q 0 G G g 0 g summand of T 0 ↑G Q . Now T is a summand of T ↑Q ↓Q = g∈[Q\G/Q] ( (T ↓Q ∩Q )) ↑Q∩ gQ

and hence a summand of some ( g(T 0 ↓Qg ∩Q )) ↑Q Q∩ gQ . For this element g we deduce that g U is Q ∩ Q-projective and by minimality of Q we have Q = Q ∩ gQ and g ∈ NG (Q). Now T is a summand of gT 0 , and since both modules are indecomposable we have T = gT 0 . A minimal subgroup Q of G relative to which the indecomposable module U is projective is called a vertex of U , and it is defined up to conjugacy in G. An RQ-module T for which U is a summand of T ↑G Q is called a source of U , and given the vertex Q it is defined up to conjugacy by elements of NG (T ). We write vtx(U ) to denote some vertex of U . (11.23) COROLLARY. Let R be a field or a complete discrete valuation ring, and let U be an indecomposable RG-module. If T is an indecomposable module for which U G is a summand of T ↑G Q where Q is a vertex of U , then T is a summand of U ↓Q . We record some immediate properties of the vertex of a module. (11.24) PROPOSITION. Let R be a field of characteristic p or a complete discrete valuation ring with residue field of characteristic p. (1) The vertex of every indecomposable module is a p-group. (2) An indecomposable module is projective if and only if its vertex is 1. (3) A vertex of the trivial module R is a Sylow p-subgroup of G. Proof. (1) We know from ? that every module is projective relative to a Sylow p-subgroup, and so vertices must be p-groups. (2) A module is projective if and only if it is projective relative to 1, which is the case for an indecomposable module if and only if 1 is a vertex. (3) Let Q be a vertex of R and P a Sylow p-subgroup of G containing Q. Then R is L G G G P a summand of R ↑G Q , so R ↓P is a summand of R ↑Q ↓P = g∈[P \G/Q] R ↑P ∩ gQ and hence is a summand of R ↑P P ∩ gQ for some g ∈ G. We claim that for every subgroup H ≤ P , P R ↑H is an indecomposable RP -module. From this it will follow that R = R ↑P P ∩ gQ and that Q = P . The only simple RP -module is the residue field k with the trivial action ∼ (in case R is a field already, R = k), and HomRP (R ↑P H , k) = HomRH (R, k) = k is a space of dimension 1. This means that R ↑P H has a unique simple quotient, and hence is indecomposable.

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(11.25) LEMMA. Let Q be a subgroup of G and L a subgroup with L ⊇ NG (Q). Let g ∈ G − L. Then L ∩ gQ is not conjugate in L to Q. Proof. Suppose that x(L ∩ gQ) = Q for some x ∈ L. Then xg ∈ NG (Q) ⊆ L. Since x ∈ L it follows that g ∈ L.

xg

Q = Q so that

(11.26) THEOREM (Green correspondence). Let R be a field of characteristic p or a complete discrete valuation ring with residue field of characteristic p. Let Q be a p-subgroup of G and L a subgroup of G which contains the normalizer NG (Q). (1) Let U be an indecomposable RG-module with vertex Q. Then in any decomposition of U ↓G L as a direct sum of indecomposable modules there is a unique indecomposable summand f (U ) with vertex Q. Writing U ↓G L = f (U ) ⊕ X, each summand of X is projective relative to a subgroup of the form L ∩ gQ where g ∈ G − L. (2) Let V be an indecomposable RL-module with vertex Q. Then in any decomposition of V ↑G L as a direct sum of indecomposable modules there is a unique indecomposable summand g(V ) with vertex Q. Writing V ↑G L = g(V ) ⊕ Y , each summand of Y is projective relative to a subgroup of the form Q ∩ gQ where g ∈ G − L. (3) In the notation of parts (1) and (2) we have gf (U ) ∼ = U and f g(V ) ∼ = V. Proof. We will prove statement (2) before statement (1). Let V be an indecomposable RL-module with vertex Q. Step 1. We show that in any decomposition as a direct sum of indecomposable modG ules, V ↑G L ↓L has a unique summand with vertex Q, the other summands being projective relative to subgroups of the form L ∩ gQ with g 6∈ L. To show this, let T be a source for ∼ V , so that T ↑L Q = V ⊕ Z for some RL-module Z. Put G 0 V ↑G L ↓L = V ⊕ V , G 0 Z ↑G L ↓L = Z ⊕ Z

for certain RL-modules V 0 and Z 0 . Then G G G G G T ↑G Q ↓L = V ↑L ↓L ⊕Z ↑L ↓L

= V ⊕ V 0 ⊕ Z ⊕ Z0 M = ( g(T ↓Lg ∩Q )) ↑L L∩ gQ . g∈[L\G/Q]

There is one summand in the last direct sum with g ∈ L and it is isomorphic to T ↑L Q= g V ⊕ Z. The remaining summands are all induced from subgroups L ∩ Q with g 6∈ L, and it follows that all indecomposable summands of V 0 and Z 0 are projective relative to these subgroups. This in particular implies the assertion we have to prove in this step. Step 2. We show that in any decomposition as a direct sum of indecomposable modules, V ↑G L has a unique indecomposable summand with vertex Q and that the remaining

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summands are projective relative to subgroups of the form L ∩ gQ where g 6∈ L. To show this, write V ↑G L as a direct sum of indecomposable modules and pick an indecomposable summand U for which U ↓G L has V as a summand. This summand U must have vertex Q; for it is projective relative to Q, since V is, and if U were projective relative to a smaller group then V would be also, contradicting the fact that Q is a vertex of V . This shows that the direct sum decomposition of V ↑G L has at least one summand with vertex Q. 0 G 0 Let U be another summand of V ↑L . Then U 0 ↓G L must be a summand of V , in the notation of Step 1, and every indecomposable summand of U 0 ↓G L is projective relative to y 0 a subgroup L ∩ Q with y 6∈ L. Since U is a summand of T ↑G Q it is projective relative 0 to Q, and hence has a vertex Q which is a subgroup of Q. Since L ⊇ Q0 it follows that 0 0 U 0 ↓G L has an indecomposable summand which on restriction to Q has a source of U as a summand, and so Q0 is a vertex of this summand. It follows that some L-conjugate of Q0 must be contained in one of the subgroups L ∩ yQ with y 6∈ L. In other words xQ0 ⊆ L ∩ yQ −1 for some x ∈ L. Thus Q0 ⊆ x yQ where g = x−1 y 6∈ L. This shows that Q0 ⊆ Q ∩ gQ and completes the proof of assertion (2) of this theorem. Step 3. We establish assertion (1). Suppose that U is an indecomposable RG-module with vertex Q. Letting T be a source of U , there is an indecomposable summand V of G G T ↑L Q for which U is a summand of V ↑L . This is because U is a summand of T ↑Q = G (T ↑L Q ) ↑L . This RG-module V must have vertex Q, since it is projective relative to Q, and if it were projective relative to a smaller subgroup then so would U be. Now U ↓ G L is G G a direct summand of V ↑L ↓L , and by Step 1 this has just one direct summand with vertex Q, namely V . In fact U ↓G L must have an indecomposable summand which on further restriction to Q has T as a summand, and this summand has vertex Q. It follows that this summand must be isomorphic to V , and in any expression for U ↓G L as a direct sum of indecomposable modules, one summand is isomorphic to V and the rest are projective relative to subgroups of the form L ∩ gQ with g 6∈ L. This completes the proof of assertion (1) of the theorem. Step 4. The final assertion of the theorem follows from the first two and the fact that G G G U is isomorphic to a summand of U ↓G L ↑L and V is isomorphic to a summand of V ↑L ↓L . THEOREM (Burry-Carlson-Puig). THEOREM. Maps. Reiten’s construction of the indecomposable modules. Green Peacock. Examples of Green correspondence in action. Green’s indecomposability theorem. Green correspondence is one of the main tools available to us in understanding the indecomosable representations of a group algebra. It shows that to some degree these representations are determined by the representations of the normalizers of p-subgroups.

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This approach has been the principal area of development in modular representation theory of finite groups in recent times. We will see it in action in the next chapter, which has to do with blocks. We mention also another technique which has been used to give positive information about indecomposable representations of finite-dimesnional algebras, namely the theory of almost split sequences. This has its context in the representation theory of finitedimensional algebras in general, not just group algebras, and is described in [Ben], and in more detail in [ARS]. In the context of group algebras this approach was used by Erdmann [Erd] in classifying the structure of (blocks of) group algebras of tame representation type. Exercises for Section 11. We will assume throughout these exercises that the ground ring R is either a complete discrete valuation ring, or a field k, so that the Krull-Schmidt theorem holds. 1. Write out proofs of the following assertions. They refer to RG-modules U and W , subgroups H ≤ K ≤ G and J ≤ G, and a RK-module V . (a) If U is H-projective then U is K-projective. g (b) If U is H-projective and W is an indecomposable summand of U ↓G J then W is J ∩ Hprojective for some element g ∈ G. Deduce that there is a vertex of W which is contained in a subgroup J ∩ gH. (c) If U is a summand of V ↑G K and V is H-projective then U is H-projective. (d) For any g ∈ G, U is H-projective if and only if gU is gH-projective. (e) If U is H-projective and W is any RG-module then U ⊗ W is H-projective. 2. Consider an even-dimensional indecomposable k[C2 × C2 ]-module Ef,n where k is a field of characteristic 2, and suppose that f 6= 0, ∞. Define umn+1 = −amn−1 umn−1 − · · · − a1 u2 − a0 u1 and let η : Ef,n → Ef,n be the linear map specified by η(ui ) = ui+1 , η(vi ) = vi+1 where 1 ≤ i ≤ mn. (1) Show that η ∈ Endk[C2 ×C2 ] (Ef,n ). (2) Show that the subalgebra khηi of Endk[C2 ×C2 ] (Ef,n ) generated by η is isomorphic to k[X]/(f n ). (3) Show that khηi + Homk[C2 ×C2 ] (Ef,n , Soc(Ef,n )) = Endk[C2 ×C2 ] (Ef,n ). (4) Deduce that Ef,n is an indecomposable k[C2 × C2 ]-module. 3. Let G = hai × hbi = C2 × C2 and let U be an even-dimensional indecomposable kG-module where k is a field of characteristic 2. (1) Prove that if U 6∼ = k then Rad(U ) = Soc(U ). = kG and G 6∼ Throughout the rest of this question, suppose that multiplication by a − 1 induces an isomorphism U/ Rad(U ) → Soc(U ). (2) Show that there is an action of the polynomial ring k[X] on U so that X(a − 1)u = (b − 1)u = (a − 1)Xu for all u ∈ U . Show that U ∼ = khai ⊗k (U/ Rad(U )) as khai ⊗k K[X]modules.

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(3) Show that invariant subspaces of U as a khai ⊗k K[X]-module are also invariant subspaces of U as a kG-module. Show that U/ Rad(U ) is an indecomposable k[X]-module and deduce that U/ Rad(U ) ∼ = k[X]/(f n ) as k[X]-modules for some irreducible polynomial f and integer n. (4) Prove that U ∼ = Ef,n as kG-modules. 4. Let k be a field of characteristic p and suppose that U is a kG-module with the property that for every proper subgroup H < G, U ↓G H is a projective kH-module. (1) Show that if G is a cyclic p-group then U must be a projective kG-module. (2) Show by example that if G = C2 × C2 is the Klein four-group and p = 2 then U need not be a projective kG-module. 5. Suppose that U is an RG-module which is Q-projective and that U ↓G Q has a summand which is not projective relative to any proper subgroup of Q. Show that Q is a vertex of U . 6. Let H be a subgroup of G and U an indecomposable RG-module which is a direct summand of R ↑G H . Show that the source of U is the trivial module (for the subgroup which is the vertex of U ). [Because of this, the indecomposable summands of permutation modules (over a field) are sometimes called trivial source modules.] 7. Suppose that Q is a vertex of an indecomposable RG-module U and that H is a subgroup of G which contains Q. (a) For each subgroup Q0 ⊆ H which is conjugate in G to Q, show that U ↓G H has an 0 G indecomposable summand with vertex Q . Deduce that if U ↓H is indecomposable then subgroups of H which are conjugate in G to Q are all conjugate in H. (b) Show that there is an indecomposable RH module V with vertex Q so that U is a direct summand of V ↑G H. 8. Suppose that H is a subgroup of G and that V is an indecomposable RH-module with vertex Q, where Q ≤ H. Show that V ↑G H has an indecomposable direct summand with vertex Q. Show that for every p-subgroup Q of G there is an indecomposable RGmodule with vertex Q.

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12. Blocks Let (F, R, k) be a p-modular system and G a finite group. Thus R is a discrete valuation ring with field of fractions F of characteristic zero and residue field k of characteristic p. We will define a p-block of G in a moment, but whatever it is, it determines and is determined by any of the following data: • an equivalence class of kG-modules, • an equivalence class of indecomposable kG-modules, • an equivalence class of simple kG-modules, • an equivalence class of RG-modules, • a primitive central idempotent in kG or RG, • an equivalence class of primitive idempotents in kG or RG, • an equivalence class of F G-modules, • an indecomposable 2-sided direct summand of kG, • an indecomposable 2-sided direct summand of RG, • a division of the Cartan matrix of kG into block diagonal form obtained by permuting rows and permuting columns, with as many diagonal blocks as possible. In view of this, a block can be defined by specifying any of these data, and there are many possible definitions of a block. People who work with blocks often seem to have many of these definitions in mind at the same time. To fix things, we will define a block of a ring A with identity to be a primitive idempotent in the center Z(A). We start by exploring some of the equivalent properties of blocks. First we recall without proof the statement of Proposition 3.22. (12.1) PROPOSITION. Let A be a ring with identity. Decompositions A = A1 ⊕ · · · ⊕ Ar as direct sums of 2-sided ideals Ai biject with expressions 1 = e1 + · · · + er as a sum of orthogonal central idempotent elements, where ei is the identity element of Ai and Ai = Aei . The Ai are indecomposable as rings if and only if the ei are primitive central idempotent elements. If every Ai is indecomposable as a ring then the Ai , and also the primitive central idempotents ei , are uniquely determined as subsets of A, and every central idempotent can be written as a sum of certain of the ei . It makes sense to consider blocks for arbitrary algebras with identity, but the notion is particularly relevant for group algebras because here we are naturally presented with an algebra which may not be indecomposable. In a more abstract study of algebras in general we would probably make the assumption at the start that the algebra we are studying is indecomposable, and if we were to do this the notion of a block would be irrelevant.

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(12.2) PROPOSITION. Let A be a ring with identity, let 1 = e1 + · · · + en be a sum of orthogonal idempotents of Z(A) and let U be an A-module. Then U = e1 U ⊕ · · · ⊕ en U as A-modules. Thus if U is indecomposable we have ei U = U for precisely one i, and ej U = 0 for j 6= i. These summands of U satisfy HomA (ei U, ej U ) = 0 if i 6= j. This result means that each indecomposable A-module U belongs to a unique block, or lies in a unique block, namely the unique primitive central idempotent e for which eU 6= 0, and this e acts as the identity on U . More generally, we say that an A-module U belongs to e if each of its indecomposable summands belongs to e, and this is the same as requiring that eU = U . The modules which belong to a block determine that block, since in any expression for A as a direct sum of indecomposable modules the sum of the summands belonging to e is necessarily the 2-sided ideal eA. When A = RG is a group algebra, the block to which the trivial module R belongs is called the principal block. (12.3) PROPOSITION. Let G be a finite group and (F, R, k) a p-modular system in which R is complete. (1) Reduction modulo (π) gives a surjective ring homomorphism Z(RG) → Z(kG). (2) Each idempotent of Z(kG) lifts uniquely to an idempotent of Z(RG), with primitive idempotents corresponding to primitive idempotents under the lifting process. P Proof. (1) The conjugacy class sums g∼x g (i.e. the sum of all elements g conjugate to x) form a basis for Z(RG) over R, and over k they form a basis for Z(kG). Reduction modulo (π) sends one basis to the other, which proves surjectivity. (2) We have seen the lifting argument before. Since π n−1 Z(RG)/π n Z(RG) is a nilpotent ideal in Z(RG)/π n Z(RG) we may lift any idempotent en−1 + π n−1 Z(RG) to an idempotent en + π n Z(RG), thereby obtaining from any idempotent e1 + πZ(RG) ∈ Z(kG) a Cauchy sequence e1 , e2 , . . . of elements of Z(RG) whose limit is the required lift. We have also seen before that primitive idempotents correspond to primitive idempotents. In the present situation the lift of each primitive idempotent is unique since the primitive central idempotents of RG themselves are unique, so that there is only one primitive idempotent of Z(RG) which reduces to each primitive idempotent of Z(kG). Because of this, it is the same thing to study the blocks of RG and of kG since they correspond to each other under reduction modulo π. If U is a kG-module we may regard it also as an RG-module via the surjection RG → kG and if e is a block of RG with image the block e¯ ∈ Z(kG) there is no difference between the statements that U belongs to e or that U belongs e¯. We may also partition the simple F G-modules into blocks in a way consistent with the blocks for RG and kG. Regarding RG as a subset of F G, a primitive central idempotent e of RG is also a central idempotent of F G. We say that an F G module U belongs to e if eU = U . Evidently each simple F G-module belongs to a unique block. We see that if U is an RG-module and U0 ⊂ U is any R-form of U (i.e. a full RG-lattice in U ) then U0 belongs to e if and only if U belongs to e.

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Examples. 1. When A is a finite-dimensional semisimple algebra over a field, the blocks correspond to the matrix summands of A, each block being the idempotent which is the identity element of a matrix summand. Each simple module lies in its own block, and so there is only one indecomposable module in each block. We saw in Theorem 3.23 a formula in terms of characters for the primitive central idempotent in CG corresponding to each simple complex representation. 2. When G is a p-group and k is a field of characteristic p the regular representation kG is indecomposable and the identity element is a block. There is only one block in this situation and block theory does nothing for us if we are interested only in representations of p-groups in characteristic p. 3. We have seen at the end of Section 9 that when we have a block of defect zero for G over a splitting p-modular system (F, R, k) there is a 2-sided direct summand of RG which is isomorphic to a matrix algebra Mn (R), and also a matrix summand Mn (k) of kG which is the reduction modulo π of the summand of RG. There is a unique simple kG-module in this block, and it is projective. It lifts to a unique RG-lattice, and all RG-sublattices of it are isomorphic to it. A key fact which we used in identifying the features of this situation is that the primitive central idempotent of CG corresponding to the matrix summand in fact lies in RG. All this explains why we made reference to a ‘block’ in that situation, but not why we used the term ‘defect zero’. This too will soon be explained. 4. When G = S3 in characteristic 2 there are two blocks, since the simple module of degree 2 is a block of defect zero, and the only other simple module is the trivial module, which lies in the principal block. The projective cover of the trivial module as an RGmodule has character equal to the sum of the characters of the trivial representation and the sign representation, and so the principal block idempotent in RG acts as the identity on this, meaning that they are the ordinary characters in the principal block. The other ordinary character, of degree 2, lies in the other block. In characteristic 3, S3 has only one block since there are two simple modules (trivial and sign) and the sign representation appears as a composition factor of the projective cover of the trivial module. This means that it belongs to the principal block. (12.4) LEMMA. Let e be a block of a ring A with identity. If 0 → U → V → W → 0 is a short exact sequence of A-modules then V belongs to e if and only if U and W belong to e. In other words, every submodule and factor module of a module which belongs to e also belong to e, and an extension of two modules which belong to e also belongs to e. Proof. A module belongs to e if and only if multiplication by e is an isomorphism of that module, and this property holds for V if and only if it holds for U and W .

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In what follows we characterize blocks in terms of a relation on the simple modules. We could work with modules for an algebra over a complete discrete valuation ring R, but since the simple modules are naturally defined over the residue field k we will assume that A is a finite-dimensional k-algebra. The next result is an elaboration of the observation from the last lemma that each indecomposable projective module PS lies in the same block as the simple module S, and in fact all of the composition factors of PS lie in this block. (12.5) PROPOSITION. Let A be a finite-dimensional algebra over a field k. The following are equivalent for simple A-modules S and T . (1) S and T lie in the same block. (2) There is a list of simple A-modules S = S1 , S2 , . . . , Sn = T so that Si and Si+1 are both composition factors of the same indecomposable projective module, for each i = 1, . . . , n − 1. (3) There is a list of simple A-modules S = S1 , S2 , . . . , Sn = T so that for each i = 1, . . . , n − 1, Si and Si+1 appear in a non-split short exact sequence of A-modules 0 → U → V → W → 0 with {U, W } = {Si , Si+1 }. Proof. The implications (3) ⇒ (2) ⇒ (1) are straightforward in that all the composition factors of any indecomposable module necessarily belong to the same block, and whenever we have a non-split extension of the kind which appears in condition (3), the middle module is uniserial and hence indecomposable. To show that (1) ⇒ (2) we will write S ∼ T to mean that the simple modules S and T satisfy the condition of (2), which is an equivalence relation. All the composition factors of any particular indecomposable projective module are equivalent in this sense. Suppose that S and T lie in the same block. We need to show that S ∼ T . We may write the regular representation as A = P1 ⊕ · · ·⊕ Pr ⊕ Q1 ⊕ · · ·⊕ Qs where the composition factors of every Pi are equivalent to S, and none of the composition factors of the Qj are equivalent to S. Let P = P1 ⊕ · · · ⊕ Pr and Q = Q1 ⊕ · · · ⊕ Qs so that A = P ⊕ Q. In fact every submodule of A whose composition factors are all equivalent to S is contained in P , for if U is such a submodule then P + U has the same property, and (P + U ) ∩ Q = 0 since composition factors of the intersection are both equivalent and not equivalent to S. Thus A = (P + U ) ⊕ A, and it follows that P + U = P so U ⊆ P . We show that P and Q are 2-sided ideals of A. Since they are already left ideals we only need to show that they are closed under right multiplication by elements of A. Each element x ∈ A determines a module endomorphism φ : A → A which is φ(a) = ax. Now φ(P ) is a submodule of A all of whose composition factors are equivalent to S, so φ(P ) ⊆ P . This means that P x ⊆ P , so that P is a right ideal, as we were trying to show. Similarly Q is a right ideal of A, and hence a 2-sided ideal. It follows that P is a direct sum of block ideals. The block ideal determined by S is a direct summand of P , and hence every simple module in this block is equivalent to S in the sense of condition (2). (2) ⇒ (3)

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The effect of this result is that the division of the simple A-modules into blocks can be achieved in a purely combinatorial fashion, knowing the Cartan matrix of A. The statement of the next result seems confusing because the term ‘block’ is used in two different ways. It is probably the origin of the use of the term in representation theory. (12.6) COROLLARY. Let A be a finite-dimensional algebra over a field k. On listing the simple A-modules so that modules in each block occur together, the Cartan matrix of A has a block diagonal form, with one block matrix for each block of the group. Up to permutation of simple modules within blocks and permutation of the blocks, this is the unique decomposition of the Cartan matrix into block diagonal form with the maximum number of block matrices. Proof. Given any matrix we may define an equivalence relation on the set of rows and columns of the matrix by requiring that a row be equivalent to a column if and only if the entry in that row and column is non-zero, and extending this by transitivity to an equivalence relation. Now if we order the rows and columns so that the rows and columns in each equivalence class come together, the matrix is in block form, and this is the unique expression with the maximal number of blocks (up to permutation of the blocks and permutation of rows and columns within a block). The last result implies that this equivalence relation coincides with the division of modules into blocks. Example. Suppose that G = K o H where K has order prime to p and H is a pgroup. In other words, G has a normal p-complement and is termed p-nilpotent. We saw in Theorem 8.10 that this is precisely the situation in which each indecomposable projective module has only one isomorphism type of composition factor. Another way of expressing this is to say that the Cartan matrix is diagonal. In view of the last results we see that we have characterized the groups for which each p-block contains just one simple module over a field of characteristic p as being the p-nilpotent groups. We can describe the primitive central idempotents explicitly in this situation. (12.7) PROPOSITION. Let G = K o H be a p-nilpotent group, where K has order prime to p and H is a p-group. Let k be a field of characteristic p. Each block of kG lies in kK, and is the sum of a G-conjugacy class of blocks of kK. Proof. Observe that if 1 = e1 + · · · + en is the sum of blocks of kK then for each i and g ∈ G the conjugate gei g −1 is also a block of kK. For this we verify that this element is idempotent, and also that it is central in kK, which is so since if x ∈ K then xgei g −1 = g(g −1 xg)ei g −1 = gei (g −1 xg)g −1 = gei g −1 x. Furthermore gei g −1 is primitive in Z(kK) since if it were the sum of two orthogonal central idempotents, on conjugating back by g −1 we would be able to deduce that ei is not primitive either. The blocks of kK are uniquely determined, and it follows that gei g −1 = ej for some j. Thus G permutes the blocks of kK.

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P For each fixed i the element f = e=gei g −1 e ∈ Z(kK) is idempotent. It is in fact P central in kG since if x ∈ G then xf x−1 = e=gei g −1 xex−1 = f , the sum again being over the elements in the G-orbit of ei . We now show that f is primitive in Z(kG). Suppose instead that f = f1 + f2 is a sum of orthogonal idempotents in Z(kG). Then there are non-isomorphic simple kG-modules U1 and U2 with f1 U1 = U1 and f2 U2 = U2 . Since f f1 = f1 and f f2 = f2 we have f U1 = U1 and f U2 = U2 . a ∼ a By Clifford’s theorem U1 ↓G K = S1 ⊕· · ·⊕St for some integer a, where S1 , . . . , St are Gconjugate simple kK-modules. Since f ∈ kK we have f Si = Si for all i, and this identifies these modules exactly as the G-orbit of simple kK modules associated to f . By a similar b ∼ b argument U2 ↓G K = S1 ⊕ · · · ⊕ St for some integer b, where the Si are the same modules. Since K consists of the p-regular elements of G it follows that the Brauer characters of U1 and U2 are scalar multiples of one another. Since Brauer characters of non-isomorphic simple modules are linearly independent we deduce that U1 ∼ = U2 , a contradiction. This shows that f is primitive in Z(kG). We may see the phenomenon described in the last result in many examples, of which the smallest non-trivial one is G = S3 = K oH where K = h(1, 2, 3)i and H = h(1, 2)i. Let k = F4 . The blocks of kK are e1 = () + (1, 2, 3) + (1, 3, 2), e2 = () + ω(1, 2, 3) + ω 2 (1, 3, 2) and e3 = () + ω 2 (1, 2, 3) + ω(1, 3, 2) where ω is a primitive cube root of 1 in F4 . In the action of G on these there are two orbits, namely {e1 } and {e2 , e3 }. The blocks of kG are e1 and e2 + e3 = (1, 2, 3) + (1, 3, 2).

The defect of a block There is more than one approach to the definition of a defect group of a block. We will start with a module-theoretic approach, and after that relate it to a ring-theoretic approach. There are two ways to obtain the regular representation from the group ring RG. One is to regard RG as a left RG-module via multiplication from the left, but there is another left action of G in which an element g ∈ G acts by multiplication from the right by g −1 . The module we obtain in this way is isomorphic to the regular representation obtained via left multiplication. Because these two actions commute with each other we may combine them, and regard kG as a representation of G × G with an action given by (g1 , g2 )x = g1 xg2−1

where g1 , g2 ∈ G, x ∈ kG.

It will be important to consider the diagonal embedding of δ : G → G × G specified by δ(g) = (g, g). Via this embedding we obtain yet another action of G on RG, which is the action given by conjugation: g · x = gxg −1 .

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PROPOSITION. (1) The submodules of RG, regarded as a module for R[G × G], are precisely the 2-sided ideals of RG. (2) Any decomposition of RG as a direct sum of indecomposable R[G × G]-modules is a decomposition as a direct sum of blocks. (3) Regarded as a representation of G × G, RG is a permutation module in which the stabilizer of 1 is δ(G). Thus RG ∼ = R ↑G×G δ(G) . COROLLARY. Let R be a discrete valuation ring with residue field of characteristic p (or a field of characteristic p) and let G be a finite group. Let e be a block of RG. Then regarded as a R[G × G]-module the summand eRG has a vertex of the form δ(D) where D is a p-subgroup of G. Such a subgroup D is defined up to conjugacy in G. We define a subgroup D of G to be a defect group of the block e if δ(D) is a vertex of eRG. It is defined up to conjugacy in G. If |D| = pd we say that d is the defect of e, but often we abuse this terminology and say simply that the defect of the block is D. According to these definitions, a block of defect 0 is one whose defect group is the identity subgroup. It is not immediately apparent that this definition of a block of defect zero coincides with the previous one, and this will have to be proved. The module-theoretic approach to the defect group of a block, which was pioneered by J.A. Green, allows us to obtain more specific information about the kinds of groups which can be defect groups. THEOREM (Green). Let e be a p-block of G with defect group D, and let P be a Sylow p-subgroup of G which contains D. Then D = P ∩ gP for some element g ∈ CG (D). Proof. Let P be a Sylow p-subgroup containing D. We consider G×G G×G RG ↓G×G P ×P =↑δ(G) ↓P ×P M =

δ(G)

×P ( y(R ↓(P ×P )y ∩δ(G) )) ↑P (P ×P )∩ yδ(G)

y∈[P ×P \G×G/δ(G)]

=

M

y∈[P ×P \G×G/δ(G)]

×P R ↑P (P ×P )∩ yδ(G) .

×P Now each R ↑P (P ×P )∩ yδ(G) is indecomposable since P × P is a p-group, as shown in the proof of part (3) of Proposition 11.24. The summand eRG of RG on restriction to P × P has a summand which on further restriction to δ(D) has a source of eRG as a summand. G×G This summand of eRG ↓P ×P also has vertex δ(D) (an argument which is familiar from the ×P proof of the Green correspondence 11.26), and must have the form R ↑P (P ×P )∩ yδ(G) for some y ∈ G×G. Thus δ(D) is conjugate in P ×P to (P ×P )∩ yδ(G), so δ(D) = z((P ×P )∩ yδ(G)) for some z ∈ P × P .

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The elements 1 × G = {(1, t) t ∈ G} form a set of coset representatives for δ(G) in G × G, and so we may assume y = (1, t) for some t ∈ G. Write z = (r, s), where r, s ∈ P . Now (P × P ) ∩ (1,t)δ(G) = {(x, tx) x ∈ P and tx ∈ P } = {(x, tx) x ∈ P ∩ P t } =

(1,t)

δ(P ∩ P t )

and δ(D) = (r,s)(1,t)δ(P ∩ P t ). The projection onto the first coordinate here equals D = −1 −1 r (P ∩ P t ) = rP ∩ rt P = P ∩ rt P since r ∈ P . At this point it almost looks as though the proof is complete, apart from the fact that rt−1 might not centralize D. Now −1 −1 −1 (1,t−1 )(r −1 ,s−1 ) δ(D) ⊆ δ(G), so that r x = t s x for all x ∈ D, and rt−1 s−1 ∈ CG (D). −1 −1 Since s ∈ P we have D = P ∩ rt s P and this completes the proof. COROLLARY. If e is a p-block of G with defect group D then D = Op (NG (D)) ⊇ Op (G), where Op (G) denotes the largest normal p-subgroup of G. Proof. We start by observing that OP (G) is the intersection of the Sylow p-subgroups of G; for the intersection of the Sylow p-subgroups is a normal p-subgroup since the Sylow p-subgroups are closed under conjugation, and on the other hand Op (G) ⊆ P for some Sylow p-subgroup P , and hence g(Op (G)) = Op (G) ⊆ gP for each element g ∈ G. Since g P accounts for all Sylow p-subgroups by Sylow’s theorem it follows that Op (G) is contained in their intersection. It follows immediately that D ⊇ Op (G) since D is the intersection of two Sylow p-subgroups and hence contains the intersection of all Sylow p-subgroups. To prove that D = Op (NG (D)), let P be a Sylow p-subgroup of G which contains a Sylow p-subgroup of NG (D). Such a P necessarily has the property that P ∩ NG (D) is a Sylow p-subgroup of NG (D). Now D = P ∩ gP for some g ∈ CG (D) and so in particular g ∈ NG (D). Thus gP ∩ NG (D) = g(P ∩ NG (D)) is also a Sylow p-subgroup of NG (D), and D = (P ∩ NG (D)) ∩ g(P ∩ NG (D)) is the intersection of two Sylow p-subgroups of NG (D). Thus D ⊇ Op (NG (D)). But on the other hand D is a normal p-subgroup of NG (D), and so is contained in Op (NG (D)). Thus we have equality. The condition on a subgroup D of G that D = Op NG (D) is quite restrictive. Such subgroups have been called p-radical subgroups by some authors in recent years, and they are also called p-stubborn subgroups. They play an important role in many questions about groups to do with topology, for example in studying the properties of classifying spaces and in studying the topological properties of the partially-ordered set of p-subgroups of G. The terminology p-radical comes from the fact that when G is a finite group of Lie type in characteristic p the p-radical subgroups are precisely the unipotent radicals of parabolic subgroups. Thus the definition of p-radical subgroup extends this notion to all finite groups. The name p-stubborn refers to the fact that in certain calculations with the partially-ordered set of p-subgroups of G, many subgroups can be ignored, but the

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p-stubborn ones cannot. The reader should be warned that the term ‘p-radical’ has also been used in some different senses, one of which is described in the book by Feit [Fei]. It is immediate that Sylow p-subgroups of a group are p-radical, as is Op (G). An exercise in group theory (presented at the end of this section) shows that other p-radical subgroups must lie between these two extremes. It is always the case that G has a p-block with defect group a Sylow p-subgroup (the principal block, for example) but it need not happen that Op (G) is the defect group of any block. We now describe another context where p-radical subgroups arise, namely Alperin’s weight conjecture. This conjecture and its refinements due to Dade have been a principal motivating force behind much research in modular representatin theory of finite groups in recent years. Let k be a splitting field for G in characteristic p. Alperin defines a weight to be a pair (H, V ) where H is a p-subgroup of G and V is a simple projective k[NG (H)/H]-module, that is, a block of defect zero for NG (H)/H. We identify two pairs (H, V ), (H1 , V1 ) if H and H1 are conjugate, and after transporting the action on V via conjugation, V and V1 are isomorphic. Thus the condition that (H, V ) = (H1 , V1 ) is that H1 = gH, and V1 ∼ = gV . CONJECTURE (Alperin). For every finite group G, the number of weights for G in characteristic p equals the number of simple kG-modules. This number is also equal to the number of p-regular conjugacy classes of G, and from this point of view is easy to determine. What is more mysterious is the number of blocks of a group, and one of the remarkable things about this conjuecture is that it makes a connection between two apparently different sets of objects, one of them easy to compute, the other more difficult. When (H, V ) is a weight the fact that V is a block of defect zero for N (H)/H implies that Op (NG (H)/H) = 1. Since Op (NG (H)/H) = Op (NG (H))/H this is equivalent to Op (NG (H)) = H, and so H must be a p-radical subgroup. This property enables us to list the weights for various groups. For example, when G = S4 and p = 2 the 2-radical subgroups are the normal subgroup V ∼ = C2 × C2 and the Sylow 2-subgroup D8 only. Since NG (V )/V ∼ = S3 has a 2-dimensional module which is a block of defect zero and NG (D8 )/D8 = 1 has the trivial module as a block of defect zero, there are two weights: (V, 2) and (D8 , 1). There are also two 2-regular conjugacy classes in G, as predicted by Alperin’s conjecture. When G = A5 a Sylow 2-subgroup is C2 × C2 and it is 2-radical, as is the identity subgroup. There remains one conjugacy class of 2-subgroups, which are cyclic of order 2, but since NG (C2 ) = C2 × C2 these are not 2-radical. Since NG (C2 × C2 ) = A4 and A4 /C2 ×C2 ∼ = C3 has order prime to 2, there are 3 weights of the form (C2 ×C2 , λ) where λ is a 1-dimensional representation of C3 . The remaining weights must have the form (1, V ) where V is a block of defect zero for G. In fact G has four 2-regular conjugacy classes and there is indeed a block of defect zero for A5 , a representation of degree 4: by ? the

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5 permutation representation with stablizer A4 decomposes as C ↑A A4 = C ⊕ U where U is a simple representation of degree 4.

Ring-theoretic methods The advantage of the module-theoretic approach we have just taken is that it immediately establishes the notion of a defect group of a block and its uniqueness up to conjugacy, using the theory of vertices and sources we have already developed. However, it is more convenient in many situations to adopt a ring-theoretic approach, which is not surprising in view of the fact that blocks are rings. We make some definitions which are appropriate to this context. Let R be a commutative ring with a 1 and G a group. We define an interior Galgebra over R to be an R-algebra A together with a group homomorphsim u : G → A × where A× denotes the group of units of A. Several examples of interior G-algebras are immediately available to us. The group algebra RG is itself an interior G-algebra where u is the inclusion of G as a subset of RG. Whenever A is an interior G-agebra and φ : A → B is an algebra homomorphism (sending 1A to 1B ) then B becomes an interior G-algebra via the homomorphism φu : G → B × . Thus if B is a block of G the algebra homomorphism RG → B makes B into an interior G-algebra. Each RG-module U is determined by an algebra homomorphism RG → EndR (U ) which expresses the action of RG. In fact, to specify an action of G on U is the same as specifying the structure of an interior G-algebra on EndR (U ). Whenever we have an interior G-algebra A there arise three module actions of G on A, namely an element g ∈ G can act as left multiplication by u(g), as right multiplication by u(g −1 ), and also g can act as a 7→ ga = u(g)au(g −1). This last action is particularly important since the action is via algebra automorphisms, satisfying g(ab) = ga gb. We have seen this action before when A = EndR (U ) is the endomorphism algebra of an RG-module U , and whenever we have an action of G on an interior G-algebra, this will be the action we mean. Thus fixed points AH where H ≤ G will be taken with respect to this action, P g H as will the relative trace map trG H (a) = g∈[G/H] a when a ∈ A . PROPOSITION. Let A be an interior G-algebra over R in which 1A = trG H a for H some element a ∈ A . Let U be an A-module. Then regarded as an RG-module, U is H-projective.

Proof. The representation of A on U is given by an algebra homomorphism φ : A → G EndR (U ) and now 1U = φ(1A ) = φ(trG H a) = trH φ(a). Thus by Higman’s criterion U is H-projective.

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LEMMA. Let U be an R[G × G]-module which is δ(H)-projective. Then U ↓G×G δ(G) is δ(H)-projective. G×G Proof. U is a summand of some module V ↑G×G δ(H) so U ↓δ(G) is a summand of G×G V ↑G×G δ(H) ↓δ(G) =

M

δ(G)

δ(H)

( x(V ↓δ(G)x ∩δ(H) )) ↑δ(G)∩ xδ(H) .

x∈[δ(G)\G×G/δ(H)]

In this formula we may write each x ∈ G × G as x = (a, b) = (a, a)(1, a−1b) and now δ(G) ∩ xδ(H) =

(a,a)

(δ(G) ∩

=

(a,a)

{(H, a

=

(a,a)

−1

(1,a−1 b)

δ(H)) −1 b h) h = a bh}

δ(CH (a−1 b)) ≤

(a,a)

δ(H).

G×G It follows that every summand of the decomposition of V ↑G×G δ(H) ↓δ(G) is projective relative

to a δ(G)-conjugate of δ(H), which is the same as being δ(H)-projective. Thus U ↓ G×G δ(G) is δ(H)-projective. PROPOSITION. Let e ∈ Z(RG) be a central idempotent of RG and H ≤ G. Then δ(G) eRG is projective relative to δ(H) as an R[G × G]-module if and only if e = trδ(H) α for some element α ∈ (eRG)H . Proof. Suppose first that eRG is δ(H)-projective as an R[G × G]-module. then by Lemma ? (eRG) ↓G×G δ(G) is δ(H)-projective. By Higman’s criterion there is an endomorphism δ(G)

α of eRG as a R[δ(H)]-module so that the identity morphism 1eRG = trδ(H) α. Now e = 1eRG (e) δ(G)

= (trδ(H) α)(e) X g α(e) = g∈[G/H]

=

X

(g, g)(α(g −1, g −1 )e))

g∈[G/H]

=

X

g(α(g −1eg))g −1

g∈[G/H]

=

X

gα(e)g −1

g∈[G/H]

= trG H (α(e)).

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δ(G)

Conversely, suppose e = trδ(H) α for some α ∈ (eRG)δ(H) . Thus hαh−1 = α and P e = g∈[G/H] gαg −1 . Now φ : eRG → eRG specified by φ(x) = αx is an endomorphism of R[δ(H)]=modules, since φ((h, h)x) = αhxh−1 = h(h−1 αh)xh−1 = hαxh−1 = (h, h)φ(x). δ(G)

We claim that trδ(H) φ = 1eRG . For X

δ(G)

(trδ(H) φ)(x) =

(g, g), φ((g −1, g −1 )x)

g∈[G/H]

X

=

(g, g)φ(g −1xg)

g∈[G/H]

X

=

(g, g)αg −1xg

g∈[G/H]

X

=

gαg −1 xgg −1

g∈[G/H]

=

X

gαg −1 x

g∈[G/H]

= ex = x. G×G

This shows that eRG ↓δ (G) is δ(H)-projective, and hence since eRG is δ(G)-projective this implies that eRG is in fact δ(H)-projective. Putting the last results together we obtain a proof of the following characterization of the defect group in terms of the effect of the relative trace map on the interior G algebra ekG. This characterization could have been used as the definition of the defect group. THEOREM. Let e be a block of kG and D a p-subgroup of G. Then D is a defect D group of e if and only if D is a minimal subgroup with the property that trG D : (ekG) → (ekG)G is surjective. COROLLARY. Let e be a block of RG with defect group D. Then every ekG-module is projective relative to D. It is a fact (which we will not prove?) that every block has an indecomposable module with vertex exactly the defect group D, so that the defect group may be characterized as the unique maximal vertex of modules in the block.

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COROLLARY. The defect groups of the principal block are the Sylow p-subgroups of G. Proof. A defect group is a p-subgroup of G, and it must contain a vertex of the trivial module, which is a Sylow p-subgroup. We are now in a position to show that our previous use of the term ‘block of defect 0’ is consistent with the definitions of this section. COROLLARY. Assume that k is a splitting field for G. A block of kG has defect zero in the sense of this section if and only if it has defect zero in the sense of Section 9, which can be characterized as saying that the block is a matrix algebra over k. Proof. If the block has defect zero in the sense of this section its defect group is 1 and every module in the block is 1-projective, or in other words projective. Thus the block has a simple projective module and from the characterizations of Section 9 the block has defect zero in the sense of that Section. Conversely, suppose the block has defect zero in the sense of Section 9, so that ekG is a matrix algebra over k. We will show that ekG is projective as a k[G × G]-module, and will make use of the isomorphism k[G × G] ∼ = kG ⊗k kG. Let ∗ : kG → kG be the algebra anti-isomorphism which sends each group element g to its inverse, so that P P ( λg g)∗ = λg g −1 . An element x in the second kG factor in the tensor product acts on ekG as right multiplication by x∗ , and it follows that e ⊗ e∗ acts as the identity on ekG. Now e ⊗ e∗ is a central idempotent in kG ⊗k kG which generates the 2-sided ideal ekG ⊗k e∗ kG = ekG ⊗k (ekG)∗ . This is the tensor product of two matrix algebras, since the image of a matrix algebra under an anti-isomorphism is a matrix algebra. Such a tensor product is again a matrix algebra, for if Eij and Fkl are two matrix algebra bases (consisting of the matrices which are non-zero in only one place, where the entry is 1), then the tensors Eij ⊗k Fkl are a basis for the tensor product which multiply together in the manner of a matrix algebra basis. We see that the block ekG ⊗k e∗ kG of kG ⊗k kG is semisimple, and so ekG is a projective k[G × G]-module.

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The Brauer morphism and correspondence of blocks In this section we will work over a field k of characteristic p. LEMMA. Let e be an idempotent in a ring A. (1) (Rosenberg’s lemma) Suppose that eAe is a local ring and that e ∈

P

j Ij

where the

Ij are members of a family of ideals of A (right, left or 2-sided). Then e ∈ Ij for some j.

(2) The idempotent e is primitive in A if and only if e is the only idempotent in eAe. Thus if A is a finite-dimensional algebra over a field, eAe is a local ring if and only if e is a primitive idempotent. Proof. (1) If e 6∈ Ij for all j then every Ij is contained in the maximal ideal of eAe P and so e 6∈ j Ij since e does not lie in the maximal ideal.

(2) If e = e1 + e2 is a sum of orthogonal idempotents then eei = ei for each i and

so ei ∈ eAe. Thus e is not the only idempotent of eAe. Conversely, if e0 ∈ eAe then e = e0 + (e − e0 ) is a sum of orthogonal idempotents, and e is not primitive. The final

statement about the situation for a finite-dimensional algebra is Proposition 11.4.

When U is a kG-module and K ≤ H are subgroups of G we have the inclusion of

H K fixed points resH → U K and the relative trace map trH → U H . There is also K : U K : U

a map cg : U H → U

g

H

for each g ∈ G specified by cg (x) = gx.

LEMMA. (1) Let A be a G-algebra and H a subgroup of G. Let a ∈ AG and x ∈ AH . Then

G G G G H (trG H x)a = trH (xa) and a(trH x) = trH (ax). It follows that the image of trH : A →

AG is a 2-sided ideal.

(2) Let H, K be subgroups of G and U a kG-module. Then G resG H trK =

X

K trH H∩ gK cg resH g ∩K .

g∈[H\G/K]

Equivalently, if x ∈ U K then trG K (x) =

X

g∈[H\G/K]

trH H∩ gK (gx).

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Proof. (1) (trG H x)a = (

X

172

g

x)a

g∈[G/H]

=

X

g

xa

g∈[G/H]

=

X

g g

x a

g∈[G/H]

=

X

g

(xa)

g∈[G/H]

= trG H (xa). The rest of the argument is clear from this. (2) Writing G = Hg1 K ∪ · · · ∪ Hgn K as a disjoint union of double cosets, so that g1 , . . . , gn are double coset representatives, we have G/K = Hg1 K/K ∪ · · · ∪ Hgn K/K. Regarding this as an equality of H-sets, each Hgi K/K is permuted transitively by H, and StabH (gi K) = H ∩ giK. Thus Hgi K/K ∼ = H/(H ∩ giK) as H-sets, so that as h ranges through a set of left coset representatives for H ∩ giK in H, so hgi ranges through a set of left coset representatives for K in Hgi K. Applying the sum of these to x gives P H h∈[H/(H∩ giK)] hgi x = trH∩ giK (gi x). Summing over the (H, K)-double cosets in G gives the right side of the equation to be established, and it equals the left side because each left coset representative of K in G is applied to x just once in this sum. We now combine these constructions and define the Brauer quotient X K U (H) = U H / trH K (U ). K
G We also define the Brauer morphism BrG → U (H) to be the composite H :U resG

H H U G −→U → U (H)

where the latter map is the quotient homomorphism. In this generality U G is simply a vector space and U (H) has the structure of a k[NG (H)]-module. The image of BrG H lies in the fixed points under the action of NG (H). If U happens also to be a G-algebra then P H K H by the above lemma, the Brauer quotient U (H) is an K
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LEMMA. Let H and J be subgroups of G, let U be a kG-module and let a ∈ U J . If BrH (trG J (a)) 6= 0 then H is conjugate to a subgroup of J . P G H G Proof. BrH (trG J (a)) is the image in U (J ) of resH trJ (a) = g∈[H\G/J ] trH∩ gJ (ga). If this is not zero in U (J ) then for some term in the sum, H ∩ gJ must not be a proper subgroup of H, or in other words H ⊆ gJ . The next result provides two more characterizations of the defect group of a block. PROPOSITION. Let e be a block of kG. The following are equivalent. (1) e has defect group D. D (2) e = trG and BrG D (e) 6= 0. D (a) for some element a ∈ (kG) (3) D is up to conjugacy the unique maximal subgroup of G such that BrG D (e) 6= 0. Proof. (1) ⇒ (2) If e has defect group D then D is minimal among groups for which D G e = trG D (a) for some a ∈ (kG) . Thus certainly e = trD (a). Suppose that BrD (e) = 0. P P K K then e ∈ K
X

trD K (uK ))

K
= trG D (a

X

trD K (uK ))

K
=

trG D

=

X

X

trD K (auK )

K
trG K (auK ).

K
P K K Thus e ∈ K
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LEMMA. Let Ω be an H-set and kΩ the corresponding permutation module. Then (kΩ)H has as a basis the elements which are sums of H-orbits on Ω. When H is a p-group P H K spanned by the orbit sums where the orbit has size larger than 1. K
Proof. In the conjugation action kG is a permutation module and the set of fixed points of H on G is CG (H). Thus the Brauer morphism may be identified as projection P K onto the first factor in the decomposition kG = k[CG (H)] ⊕ K
(kG)G k

G BrG K : (kG)

resG H

−→

resG K

−→

(kG)H −→   H yresK

(kG)K

−→

kG(H)

=

kG(K) =

kCG (H)  ι y

kCG (K)

G where ι is the inclusion of the centralizer group rings, so that BrG K is the composite ι ◦ BrH . G G We see from this that if BrG H (x) 6= 0 for some x ∈ (kG) then BrK (x) 6= 0 for every K ≤ G G H, since BrH (x) equals BrK (x) with its support truncated to CG (H). This observation

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provides a strengthening of part of Proposition ?, that if K is any subgroup of a defect group D of a block e of kG, then BrK (e) 6= 0. We illustrate with G = S3 and k = F2 where we have blocks e1 = ()+(1, 2, 3)+(1, 3, 2) and e2 = (1, 2, 3) + (1, 3, 2). When H = h(1, 2)i we have CG (H) = H and BrG H (e1 ) = (), G G G BrH (e2 ) = 0. We also have Br1 (e1 ) = e1 and Br1 (e2 ) = e2 showing that h(1, 2)i and 1 are (respectively) the largest subgroups H (up to conjugacy) for which Br H (e1 ) 6= 0 and BrH (e2 ) 6= 0. These subgroups are the defect groups of the blocks, as asserted by Proposition ?. COROLLARY. Let e be a block of kG with defect group D, and suppose g ∈ G is an element which centralizes a p-power element of G which does not lie in any conjugate of D. Then g does not lie in the support of e. Proof. Let x be a p-power element of G not in any conjugate of D. Then Brhxi (e) = 0 since hxi is not contained in any conjugate of D. Since the coefficients of g in e and Brhxi (e) are the same, the result follows. For example, if e is a block of defect zero corresponding to a character χ, then χ(g) ∈ (π), the maximal ideal of the discrete valuation ring in a p-modular system (F, R, k), if g commutes with any element of order p. We now define the Brauer correspondence of blocks. Since k[CG (H)]NG (H) ⊆ k[CG (H)]CG (H) = Z(k[CG (H)]) the Brauer morphism may be regarded as a map BrG H : Z(kG) → Z(k[CG (H)]) which is a ring homomorphism, and since it is obtained by truncating the support of group ring elements outside CG (H) it sends 1 to 1. It follows that if e is any central idempotent of kG then BrG H (e) is a central idempotent of k[CG (H)] in the center of NG (H), and G if 1 = e1 + · · · + en is the sum of blocks of kG then 1 = BrG H (e1 ) + · · · + BrH (en ) is a sum of orthogonal central idempotents of k[NG (H)]. If J is a subgroup of G with CG (H) ⊆ J ⊆ NG (H) and b is a block of kJ then there is a unique block e of G so that G G b is a summand of BrG H (e), in the sense that b BrH (e) = b. We write b for this block e, and call it the Brauer correspondent of b. Notice that the choice of subgroup H does not matter in the definition, since the Brauer homomorphism truncates a group ring element outside CG (H), and if H1 were another subgroup of G with the same centralizer the Brauer homomorphism with respect to H1 would be the same. PROPOSITION. Suppose that Q is a normal p-subgroup of G. Then every block of G lies in k[CG (Q)] and is the sum of a G-orbit of blocks of CG (Q). P K Proof. We show first that K
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176

S ↓G Q is semisimple and the trivial module is the only simple kQ-module). Thus if a ∈ P P −1 (kG)K and u ∈ S we have trQ u = g∈[Q/K] au = |Q : K|au = 0. K (a) · u = g∈[Q/K] gag P Q K Thus annihilates every simple kG-module and so is contained in the K
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G LEMMA. Let H be a subgroup of G and U an kG-module. Then BrG H trH = N (H) H trHG BrH → U (H)NG (H) . H :U

P H G G G Proof. BrG H trH (x) is the image in U (H) of resH trH (x) = g∈[H\G/H] trH∩ gH (gx). The only terms which contribute have H ∩ gH = H, or in other words g ∈ NG (H), so the P N (H) image equals the image of g∈[NG (H)/H] gx = trHG x. The following is a basic version of Brauer’s ‘first main theorem’.

THEOREM (Brauer’s first main theorem). Let D be a p-subgroup of G. The Brauer morphism induces a bijection between blocks of kG with defect group D and blocks of NG (D) with defect group D. Proof. Let us write N for NG (D) and let b ∈ Z(kN ) have defect D. Then b = trN D (a) P D N D K for some a ∈ (kN ) = k[CG (D)] ⊕ K
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178

The subgroups C2 have Sylow p-subgroups P as their normalizers and there are no blocks of kP with C2 as defect group, since P is a 2-group. Hence there are no blocks of kG with this defect group. This duplicates information we know from a different source, to the effect that C2 cannot be a defect group by ? because it is not O2 of its normalizer. Finally there remain the blocks of defect zero of kA5 , which have defect group 1. There is in fact just one of these, as may be seen by inspecting the character table of A5 for characters of degree divisible by 4. Exercises for Section 12. 1. Prove that if a module U lies in the principal block then so does U ∗ . 2. Use the results of Section 10 Exercises 3 5 to show that the simple group GL(3, 2) of order 168 has two blocks in characteristic 2 and two blocks in characteristic 7. 3. Suppose that Q1 and Q2 are subgroups of G which satisfy Qi = Op (NG (Qi )) for i = 1, 2, and let Ni = NG (Qi ) for each i. Prove that if N1 ⊇ N2 then Q1 ⊆ Q2 . Deduce that Qi ⊇ Op (G) for each i. LEMMA. Let U and V be A-modules which lie in different blocks of an algebra A. Then (1) HomA (U, V ) = 0, and (2) every short exact sequence of A-modules 0 → U → W → V → 0 is split. PROPOSITION. Let (F, R, k) be a complete p-modular system and G a finite group. Let the decomposition number dT S be the multiplicity of the simple kG-module S as a composition factor of the reduction modulo π of the simple F G-module T . (1) The simple F G-modules T for which dT S 6= 0 for some simple module S in a fixed block of kG constitute the simple modules in a p-block of F G-modules. (2) The simple kG-modules S for which dT S 6= 0 for some simple module T in a fixed block of F G constitute the simple modules in a block of kG-modules.

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Appendix: Discrete valuation rings Let F be a field. A valuation on F is a mapping φ : F → R≥0 such that • φ(a) = 0 if and only if a = 0, • φ(ab) = φ(a)φ(b) for all a, b ∈ F , and • φ(a + b) ≤ φ(a) + φ(b) for all a, b ∈ F . (A.1) Examples. 1. No matter what the field F is, we always have the valuation φ(a) =

n

0 if a = 0 1 otherwise.

This valuation is the trivial valuation, and we generally exclude it. 2. If F is any subfield of the complex numbers we may take φ(a) = |a|, the absolute value of a. 3. Let F = Q and pick a prime p. Every rational number a ∈ Q may be written a = rs where (r, s) = 1. We set

νp (a) =

(

∞ power to which p divides r −power to which p divides s

so that a = pνp (a)

if a = 0, if p r, otherwise,

r0 s0

where (r 0 , p) = 1 = (s0 , p). Now let λ be any real number with 0 < λ < 1 and put φ(a) = λνp (a) . Often λ is taken to be p1 , but the precise choice of λ does not affect the properties of the valuation. This valuation is called the p-adic valuation on Q. This last φ is an example of a valuation which satisfies the so-called ultrametric inequality φ(a + b) ≤ max{φ(a), φ(b)},

which in the case of this example comes down to the fact that if pn a and pn b where a, b ∈ Z, then pn (a + b). We say that φ is non-archimedean if it satisfies the ultrametric inequality. The valuations in the third example are also discrete, meaning that {φ(a) a ∈ K, a 6= 0} is an infinite cyclic group under multiplication. It is the case that discrete valuations are necessarily non-archimedean. We deduce from the axioms for a valuation that φ(1) = φ(−1) = 1. Using this we see that every valuation φ gives rise to a metric d(a, b) = φ(a − b) on the field F . We say that two valuations are equivalent if and only if the metric spaces they determine are equivalent, i.e. they give rise to the same topologies. In Example 3 above, changing the value of λ between 0 and 1 gives an equivalent valuation.

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(A.1) THEOREM (Ostrowski). Up to equivalence, the non-trivial valuations on Q are the ones just described, namely the usual absolute value and for each prime number a non-archimedean valuation. More generally, if R is a ring of algebraic integers (or more generally a Dedekind domain) with quotient field F , the non-archimedean valuations on R, up to equivalence, biject with the maximal ideals (which are the same as the non-zero prime ideals) of R. Let φ be a non-archimedean valuation on a field F . We rapidly verify that the set Rφ = {a ∈ F φ(a) ≤ 1} is a ring, called the valuation ring of φ. Any ring arising in this way is called a valuation ring, and if the valuation is discrete the ring is called a discrete valuation ring. We set Pφ = {a ∈ F φ(a) < 1}, and this is evidently an ideal of Rφ . For example, if F = Q and φ is the non-archimedean valuation corresponding to the prime p then Rφ is the localization of Z at p, and Pφ is its unique maximal ideal. (A.2) PROPOSITION. Let φ be a discrete valuation on a field F with valuation ring Rφ and valuation ideal Pφ . (1) An element a ∈ Rφ is invertible if and only if φ(a) = 1. (2) Pφ is the unique maximal ideal of Rφ , consisting of the non-invertible elements. (3) Pφ = (π) where π is any element such that φ(π) generates the value group of φ. (4) Every element of Rφ is uniquely expressible a = π ν a0 where a0 is a unit in Rφ . In this situation φ(a) = φ(π)ν . (5) The ideals of Rφ are precisely the powers Pφn = (π n ). Thus Rφ is a principal ideal domain. (6) F is the field of fractions of Rφ . Proof. The proofs are all rather straightforward. If ab = 1 where a, b ∈ Rφ then φ(ab) = φ(a)φ(b) = φ(1) = 1 and since φ(a) and φ(b) are real numbers between 0 and 1 it follows that they equal 1. Conversely, if φ(a) = 1 let b be the inverse of a in F . Since φ(b) = 1 also it follows that b ∈ Rφ and a is invertible in Rφ . This proves statement (1). Now Pφ is seen to consist of the non-invertible elements of Rφ . It is an ideal, and it follows that it is the unique maximal ideal. Defining π to be an element for which φ(π) generates the value group of φ we see that π ∈ Pφ . If a ∈ Pφ is any element then φ(a) = φ(π)ν for some ν and so φ(π −ν a) = 1. Thus π −ν a = a0 for some element a0 ∈ Rφ which is a unit, and so a = π ν a0 . This proves that Pφ = (π), and also the first statement of (4). The uniqueness of the expression in (4) comes from the fact that in any expression a = π ν a0 with a0 a unit, necessarily ν is defined by φ(a) = φ(π)ν , and then a0 is forced to be π −ν a. To prove (5), if I is any ideal of Rφ we let n be minimal so that I contains a non-zero element a with φ(a) = φ(π)n . Then by (4) we have (a) = (π n ) ⊇ I, and it follows that I = (π n ).

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As for (6), given any element a ∈ F , either a ∈ Rφ or φ(a) = φ(π)−n for some n > 0. 0 In the second case the element a0 = π n a has φ(a0 ) = 1 so a0 ∈ Rφ and now a = πan , showing that a lies in the field of fractions of Rφ in both cases. When we reduce representations from characteristic zero to positive characteristic we need to work with algebraic number fields, that is, field extensions of Q of finite degree. Let F be an algebraic number field, and R its ring of integers. We quote without proof some facts about this situation. A full account may be found in [Cas] or [FT]. A fractional ideal in F is a finitely-generated R-submodule I of F . For any such I we put I −1 = {x ∈ F xI ⊆ F }. With this definition of inverse and with a multiplication defined the same way as the multiplication of ideals, the fractional ideals form a group, whose identity is R. Every fractional ideal may be written uniquely as a product I = pa1 1 · · · pat t where p1 , . . . , pt are maximal ideals of R and the ai are non-zero integers (which may be positive or negative). Let us write νpi (I) = ai , and let 0 < λ < 1. Then for each maximal ideal p of R we obtain a discrete valuation on F by putting φ(a) = λνp (Ra) , which is called the p-adic valuation on F . (A.3) PROPOSITION. Let F be an algebraic number field with ring of algebraic integers R, and let φ be the discrete valuation on F associated to a maximal ideal p of R. Let Rp be the valuation ring of φ with maximal ideal Pp . Then Rp is the localization of R at p, and the inclusion R → Rp induces an isomorphism R/p ∼ = Rp /Pp . Proof. We assume the group structure of the set of fractional ideals. The localization of R at p is a { a, b ∈ R, b 6∈ p} b and this is clearly contained in Rp . Conversely, if ab ∈ Rp then νp (a) ≥ νp (b) and if we choose x ∈ p−νp (b) − p−νp (b)+1 then ab = ax . Now bx 6∈ p, showing that ab lies in the bx localization. The kernel of the composite R → Rp → Rp /Pp is R ∩ Pp = p. We show that this composite is surjective. We can write any element of Rp /Pp as ab + Pp where a, b ∈ R and b 6∈ p. Since p is maximal in R, R/p is a field and so there exists c ∈ R with bc−1 ∈ p. Now a a a b − ac = b (1 − bc) ∈ Pp and b + Pp = ac + Pp is the image of ac ∈ R. These observations show that we have an isomorphism R/p ∼ = Rp /Pp . Given a valuation φ on F we may form the completion Fˆ as a metric space, which contains F in a canonical way. The completion Fˆ acquires a ring structure extending that of F , and in fact Fˆ is a field. The valuation φ extends uniquely to a valuation φˆ on Fˆ , ˆ If φ is non-archimedean then so is φ, ˆ and and Fˆ is complete in the metric given by φ. ˆ with the same value group. Thus in the case of a discrete if φ is discrete then so is φ, ˆ ˆ φ = {a ∈ Fˆ φ(a) ≤ 1} with unique maximal ideal valuation we have a valuation ring R ˆ ˆ ˆ φ (π) since φ(π) Pˆφ = {a ∈ Fˆ φ(a) < 1}, and Pˆφ = R = φ(π) generates the value group. ˆ ˆ φ are exactly (We should properly write R ˆ etc, but this seems excessive.) The ideals of R φ

the powers Pˆφn .

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(A.4) LEMMA. Let φ be a discrete valuation on a field F with valuation ring Rφ . ˆ φ induces an isomorphism Rφ /P n ∼ ˆ ˆn The inclusion Rφ ,→ R φ = Rφ /Pφ for all n. ˆφ → R ˆ φ /Pˆ n . Its kernel is Proof. Consider the composite homomorphism Rφ → R φ and the desired isomorphism will follow if we can show that this homomorphism is ˆ φ we know from the construction of the completion surjective. To show this, given a ∈ R that there exists b ∈ Rφ with φ(b − a) < φ(π)n , that is, b − a ∈ Pˆφn . Now b maps to a + Pˆφn . Pφn

ˆ φ is by definition the set of equivalence classes of Cauchy sequences The completion R in Rφ . We comment that a sequence (ai ) of elements of Rφ is a Cauchy sequence if and only if for every n there exists a number N so that whenever i, j > N we have ai −aj ∈ Pφn , that is, ai ≡ aj (mod Pφn ). (A.5) LEMMA. Let φ be a discrete valuation on a field F with valuation ring Rφ , ˆ φ . Any element of R ˆ φ is uniquely expressible as a series maximal ideal Pφ and completion R a = a 0 + a1 π + a2 π 2 + · · · where the ai lie in a set of representatives S for Rφ /Pφ . ˆ φ . Since R ˆ φ /Pˆφ ∼ Proof. Let a ∈ R = Rφ /Pφ , we have a+ Pˆφ = a0 + Pˆφ for some uniquely ˆ φ . Repeating this determined a0 ∈ S. Now a − a0 ∈ Pˆφ so a = a0 + πb1 for some b1 ∈ R construction we write b1 = a1 + πb2 with a1 ∈ S uniquely determined, and in general bn = an + πbn+1 with an ∈ S uniquely determined. Now a0 , a0 + a1 π, a0 + a1 π + a2 π 2 , . . . is a Cauchy sequence in Rφ whose limit is a, and we write this limit as the infinite series. The last result, combined with Proposition A.3, provides a very good way to realize ˆ φ . For example, in the case of the p-adic valuation on Q we may take the completion R ˆ = Zp may be realized as the set of infinite S = {0, 1, . . . , p − 1}. The completion Z sequences · · · a3 a2 a1 a0 . of elements from S presented in positions to the left of a ‘point’, analogous to the decimal point (which we write on the line following American convention). Thus a0 is in the 1s position, a1 is in the ps position, a2 is in the p2 s position, and so on. Unlike decimal numbers these strings are potentially infinite to the left of the point, whereas decimal numbers are potentially infinite to the right of the point. Addition and multiplication of these strings is performed by means of the same algorithms (carrying values from one position to the next when p is exceeded, etc.) that are used with infinite decimals. Note that p-adic integers have the advantage over decimals that certain real numbers have more than one decimal representation, whereas distinct p-adic expansions always represent distinct elements of Zp . Exercises.

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1. With the description of the p-adic integers as the set of infinite sequences · · · a3 a2 a1 a0 . in positions to the left of a ‘point’, where ai ∈ {0, . . . , p − 1}, show that when p = 2 we have −1 = · · · 1111. and 1 = · · · 10101011. 3 Find the representation of the fraction 1/5 in the 2-adic integers. What fraction does · · · 1100110011. represent? ˆ of Q in the p-adic 2. Show that the p-adic rationals Qp (i.e. the completion Q valuation) may be constructed as the set of sequences · · · a3 a2 a1 a0 .a−1 a−2 · · · a−n which may be infinite to the left of the point, but must be finite to the right of the point, where ai ∈ {0, . . . , p − 1} for all i. Show that the rational numbers Q is the subset of these sequences which eventually recur.