Translations of
MATHEMATICAL Volume 2 19
Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis S. V. Kerov
American Mathematical Society
Translations of
MATHEMATICAL MONOGRAPHS Volume 2 19
Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis S . V. Kerov Translated by N. V. Tsilevich
American Mathematical Society Providence, Rhode Island
EDITORIAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. Margulis James D. Stasheff ASL Subcommittee Steffen Lempp (Chair) IMS Subcommittee Mark I. Freidlin ( C h a i r )
(Chair)
ACMMIITOTMYECKAST TEOPMST IIPEACTABJIEHMSI CMMMETPHYECKOM rPYnnb1 EE IIPMMEHEHMST B AHAJIM3E Translated from t h e Russian manuscript by N. V. Tsilevich 2000 Mathematics Subject Classification. Primary 20C30, 20P05, 22D10.
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Library of Congress Cataloging-in-Publication D a t a Kerov, S. V. (Sergei Vasil'evich), 1946-2000. [Asimptoticheskaia teoriia predstavleniia simmetricheskoi gruppy i ee primeneniia v analize. English] Asymptotic representation theory of the symmetric group and its applications in analysis / S. V. Kerov ; translated by N. Tsilevich. p. cm. - (Translations of mathematical monographs, ISSN 0065-9282 ; v. 219) Includes bibliographical references. ISBN 0-8218-3440-1 (acid-free paper) 1. Symmetry groups-Asymptotic theory. 2. Representations of groups. I. Title. 11. Series.
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Contents Foreword Publications of Sergei Kerov Chapter 0. Introduction $0. Plan of the book $1. General theory of locally semisimple algebras $2. Characters of 6,, and the Young graph $3. The Plancherel measure of 6, $4. Continuous Young diagrams in problems of analysis Chapter 1. Boundaries and Dimension Groups of Certain Graphs $1. Ergodic method $2. Combinatorial examples of branching graphs $3. The boundary and dimension group of the Kingman graph $4. Stirling triangles Chapter 2. The Boundary of the Young Graph and Macdonald Polynomials $1. Ideals of the group algebra (C[6,] $2. Characters of the infinite symmetric group $3. Generalized Macdonald polynomials and orthogonal polynomials $4. Orthogonalization of characters of the symmetric group $5. Hall-Littlewood-Macdonald symmetric polynomials 56. Duality $7. GHL-functions and generalized Legendre polynomials $8. Determinantal formulas for GHL-polynomials $9. Branching of Macdonald polynomials Chapter 3. The Plancherel Measure of the Symmetric Group $1. The typical shape of random Young diagrams $2. Gaussian limit for the Plancherel measure $3. Distribution of symmetry types of high rank tensors $4. A q-analogue of the hook walk algorithm $5. The q-analogue of the hook-length formula $6. Multiple Selberg integrals Chapter 4. Young Diagrams in Problen~sof Analysis $1. Rectangular diagrams and rational fractions $2. Continuous diagrams and R-functions $3. The Krein correspondence
vii
...
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CONTENTS
54. 55. 56. 57.
Interval shrinkage process Differential model of growth of Young diagrams Plancherel growth and semicircle diffusion Asymptotic separation of roots of orthogonal polynomials
References Comments on Kerov's Thesis. By G. OLSHANSKI Additional References
Foreword This book reproduces, without modifications, the "Doctor of Sciences" thesis of the remarkable mathematician Sergei Vassilievich Kerov (June 12, 1946-July 28, 2000). The thesis was written in 1992-1993 and defended in 1994. It is devoted mostly to analytical aspects of the asymptotic representation theory of symmetric groups and related problems of analysis. I have t o say several words about these topics, which appeared about 30 years ago. The theory itself and the name "Asymptotic representation theory" were suggested by me in the late sixties; this name refers to the whole complex of problems that lie at the interface of functional analysis, algebra, combinatorics, and probability theory, and concern the study of the behaviour of classical groups and their representations as the rank (degree) of the group tends to infinity. It is natural here to single out two classes of problems: proper asymptotic problems and problems that concern the limiting object, i.e., an infinite or infinitedimensional group which is an infinite analogue of a classical group. The principal example, which serves as a model for more difficult cases, is of course the asymptotic theory of symmetric groups and their representations. It was with this example that our activity started, just as in the 19th century the theory of finite groups and their representations had started with the symmetric groups. An early example of an asymptotic problem of the first kind was considered in the early seventies in my papers with A. Shmidt [20, 211' on the limiting joint distribution of the normalized cycle lengths of uniformly distributed random permutations. Combinatorics is here closely intertwined with probability theory. Later, many papers were devoted t o this topic, which has numerous applications (in number theory, coagulation schemes, populational genetics, etc.). This topic is considered partially in Chapter 4 of Kerov's thesis; it is also related to his later papers on the Poisson-Dirichlet measures. I formulated the dual asymptotic problem for symmetric groups in the early seventies, and it was solved in joint papers with Kerov [12, 181; this was the problem about the limiting shape of Young diagrams with the Plancherel statistics, or the typical representation of the symmetric group of high degree. From the technical point of view, it belongs to the class of "limit shape problems", which has recently become very popular. Namely, the problem asks what is the asymptotic behaviour of a configuration (for example, a Young diagram) that grows randomly but according t o certain rules. This topic is developed in Chapter 3 of the thesis, where, in particular, the central limit theorem for the Plancherel measure is proved, and numerous relations to other problems of analysis are considered. For further development see, e.g., [A.28, A.27, A.36, A.41, A.45, A.7, A.8, A.25, A.261. 'A reference such as [A.n] refers t o the additional reference list given a t the end of this volume. A reference without A refers t o the reference list in Kerov's thesis.
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A survey of a more general class of limiting shape problems can be found in [A.52] and in the papers by A. Okounkov on Schur measures, e.g., [A.38]. It was clear from the very beginning that the problem concerning the asymptotic behaviour of representations and characters is of fundamental importance. However, only much later, in the nineties, did it become clear that this problem has many relations to other topics: spectra of random matrices (see, e.g., [A.50, A.3, A.4, A.29, A.30, A.15]), free probability theory (see, e.g., [A.5-A.lO, A.48, A.49, A.14]), integrable problems and algebraic geometry (see, e.g., [A.2, A.38, A.40, A.42, A.43]), the theory of z-measures (see, e.g., [A.13, A.391). This list is far from complete. Perhaps, one relation is worth special mention. I mean the solution of the so-called Ulam's problem on the longest increasing subsequence in a sample of n independent random variables uniformly distributed on an interval (or the longest increasing subsequence in a random permutation). This problem has a long history; an important role here is played by the RSK algorithm which interprets the length of this subsequence as the length of the first row of a random Young diagram. The problem was solved completely in our paper [12], where we proved the old conjecture that the answer is 2 6 . Note that the abovementioned theorem on the limit shape of a Young diagram, which was obtained in our work and, independently, by Logan and Shepp [138],and which gives complete information on the entire monotone structure of a random sample, is, however, not sufficient t o find the asymptotics of the length of the first row, since this theorem gives only a lower bound on this length. So it required an important, significant technical observation related to Young tableaux, which was made by Kerov and completed the solution of Ulam's problem. Our proof of the general limiting shape theorem was based on a continuous analogue of the hook-length formula, estimations of probabilities, and solution of an integral equation; it was different from the proof in [138] and allowed us also to obtain estimates for the asymptotics of the maximum dimension of representations of symmetric groups. For an elementary proof of Ulanl's problem, see, e.g., [A.l]. Problems of the second kind concern the study of the infinite analogue of symmetric groups, the infinite symmetric group of finite permutations. In this class of problenls, there is a distinguished problem concerning the description of the characters of this group, which was solved by Thoma [161]by analytical techniques and which was considered from the new point of view in our paper [1.3],and solved by means of a very general ergodic method suggested in [lo] (see the first chapter of the thesis). It can be naturally reformulated as a problem on the boundary of the Young graph. In fact, this is a new class of problems which can be stated in purely analytical terms - as the description of harmonic functions of certain operators; in probabilistic terms - as the description of the Poisson or Martin boundary of as the computation of traces of AF-algebras; random walks; in algebraic terms and, finally, in combinatorial terms - as the statistics of the number of paths and central measures in graded graphs. Although the ergodic method provides a general approach, analytical difficulties are different in each specific case, and they are far from easy to overcome. Our work on the computation of the characters of the infinite symmetric group by the ergodic method [I131 was continued in papers by Okounkov [54], Olshanski and Borodin [A.12], and others, where several new proofs of Thoma's theorem were suggested and a number of other graphs were considered. The leading role here belongs to Kerov; he considered different branching -
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parameters (see Chapter 1). The results in this field that were known by 1993 are collected in Chapters 1 and 2. Later this problem was intensely studied by Kerov for other graphs and measures on them (Chapter 2). For further results about central measures and boundaries of different graphs, see, e.g., [A.24, A.21, A.22, A.23, A.19, A.201). In subsequent work, the ergodic method was repeatedly used for the description of invariant measures, characters, etc., see, e.g., [A.46, A.441. A very important general conjecture by Kerov (see [119]) on the boundary of graphs related to the Hall-Macdonald polynomials is still open. In a subsequent paper, written by Kerov together with Olshanski and Okounkov [A.34],the solution is found for a special case of Kerov's conjecture, namely, for the so-called Jack graph, a version of the Young graph with different transition probabilities. I think that the most interesting and original part of Kerov's thesis is the study of continuous Young diagrams. Already the first paper [12] on the limit shape of Young diagrams with the Plancherel distribution contained the transition from ordinary diagrams to continuous ones, and a continuous analogue of the hook-length formula. Kerov showed (see Chapter 4), first, that the limit shape of random Young diagrams with the Plancherel distribution appears naturally in a seemingly quite unrelated problem concerning separation of roots of orthogonal polynomials; and, which is even more important, that there exists a one-to-one correspondence between continuous diagrams and probability distributions that extends the correspondence between ordinary Young diagrams and their Plancherel transition probabilities; this correspondence is a nonlinear transformation of measures, which he rightly called the Markov-Krein transform. Thus Kerov linked the classical moment problem to the combinatorics of continuous Young diagrams. Ordinary diagrams correspond in this picture to discrete interlacing measures, and their Markov-Krein transform corresponds to the partial fraction expansion. Not less impressive is the relation, discovered by Kerov, of the Plancherel dynamics of continuous Young diagrams to a special solution of the Burgers equation: it turns out that the same limit shape of Young diagrams is a fixed point for this equation, and it attracts asymptotically the solutions of a certain class. And the most classical-looking result is the theorem that says that the interlacing roots of two adjacent orthogonal polynomials generate a Young diagram, and this diagram converges, as the number of the orthogonal polynomials goes t o infinity, t o the same limit shape. Kerov also found close relations of this topic to Voiculescu's free probability theory (the role of the Gaussian law in classical probability theory is played here by the senlicircle law which is related to the same limiting curve) and to combinatorics (the hook walk and the interval shrinkage process). For further results concerning the combinatorial aspects, see, e.g., [A.47, A.17, A.181. The results of Kerov on continuous diagrams and the Markov-Krein transform are set forth in more detail in his subsequent paper [A.33]. For further generalizations of the Markov-Krein transform, see [A.35, A.511. During the intervening years the results of the thesis did not become outdated; they are as interesting t o read now as they were at that time. Of course, the abovementioned progress in the study of more subtle asymptotics and relations to random matrices and free probability could not be mentioned in the thesis; however, Kerov took an active part in this new research (see the list of his publications, at the end of this Foreword).
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FOREWORD
The thesis is written very clearly and can be regarded as a handbook for mathematicians who want to work in asymptotic representation theory and obtain information on many results of the first period of its development. The introduction, which occupies almost a quarter of the whole thesis, contains all definitions and main results, so that a reader who is interested only in statements will need to read only this extensive introduction. The thesis includes only a part of the work done by Kerov up to 1994; the complete bibliography is given in this volume. I have asked G. Olshanski, A. Gnedin, and N. Tsilevich to prepare an additional list of references to the papers of subsequent years related to the topics touched upon in the thesis. Olshanski also prepared bibliographic comments, which are given at the end of the book. These comn~entscover only some of the topics that were studied in Kerov's thesis and developed later. The current state of this broad and rapidly developing area, asymptotic representation theory and its applications, demonstrates a very diverse and vivid picture. Without any doubt, Kerov's work will be avidly studied by future generations of mathematicians. As an addendum to this Foreword we reproduce an abridged version of the obituary which I wrote for the special volume "Representation theory, dynamical systems, combinatorial and algorithmic methods. VI", Zapiski Nauchn. Semin. POMI, v. 283, 2001, which was dedicated to the memory of S. Kerov; this obituary is translated by N. B. Lebedinskaya.
S. V. Kerov was a profound and original mathematician. He was gathering force not very rapidly, steadily advancing to more and more difficult problems and to the understanding of various mathematical connections. The list of his articles is not very large, but there are a number of serious papers, which will be studied and continued. His absolutely unexpected and sudden death did not allow him to complete several articles that were almost finished. I was his scientific adviser when he wrote his graduation thesis and when he worked at his Candidate's (Ph.D.) thesis.* For many years, he was my colleague and coauthor; together we elaborated ideas that I had worked on earlier, as well as those that occurred to both of us later. At a later time, he became a prominent expert on combinatorics, the theory of symmetric functions, and many other fields. He carefully examined the literature, and I often learned new information from him. His own ideas and papers, especially in the last ten years, showed his deep insight into combinatorics, analysis, and probability. Here I will try to outline his scientific path - I was a witness of much of it. In particular, I discuss in more detail our joint work, which began in the mid-seventies, continued with interruptions until the mid-eighties, and then recommenced in the late nineties. From his third year at Leningrad University, Sergei participated in the seminar on dynamical systems, representations, and algebras that I have headed since 1967. His M. Sci. thesis was devoted to flows on nilpotent and solvable manifolds. *Translatzon Editor's Note. The Russian Candidate's thesis is equivalent t o the Western Ph.D. thesis. The Russian Doctoral thesis, which this book is, is much more advanced, roughly equivalent t o "first published book".
FOREWORD
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Sergei was interested in different areas of mathematics, and from the very beginning he was very earnest and thorough in his work and study. There were many talented students (Ya. Eliashberg, Yu. Matiyasevich, and others) of the same year as Sergei, but even at that time he had a high reputation, which was based on the impression of his unceasing internal activity. His modest and dignified manner was very attractive. In 1969, he became a post-graduate student at the Department of Analysis, and I was his scientific adviser. In 1969-70, I gave a course of lectures on C*-algebras and related topics. It was a new topic to me, and I wanted to apply these techniques t o dynamical systems and representation theory. Since representation theory had gradually become the main topic, the seminar was guided in major part by problems in this theory. Sergei studied the theory of C*-algebras and the classical theory of representations of symmetric and finite groups. As a subject for his Candidate's thesis, I suggested the duality theory of *-algebras (the theory of "positive" duality). I put forward this idea generalizing the theory of Hopf algebras in 1971 and published a brief note on the geometry of states (the theory of packets) in 1972. The basic definition implied that for algebras in linear duality, niultiplication in each of them (or multiplication and comultiplication) is ail operation preserving positivity (but not multiplicativity as in Hopf algebras). The main problems were formulated, and Sergei tackled them with enthusiasm. In his Candidate's thesis, he carefully examined the finite-dimensional version, including the Plancherel duality and induction; he also studied the nontrivial commutative version (see [A.31, A.321). This theory is not yet entirely known, but Kerov's two papers on this subject are often cited. Only recently, in papers with I. Ponomarenko and S. Evdokimov, we proved that the finite-dimensional algebras in Plancherel duality are just the so-called C-algebras in algebraic combinatorics. I have no doubt that these ideas will be used in the future. There are also many relations with hypergroups, the theory of generalized shifts, quantum groups, many-valued groups, and so on. Sergei studied B. M. Levitan's book thoroughly, and we discussed further applications of the theory of positive duality to differential equations. By 1975, when his Ph.D. thesis was completed, I had drawn him t o a new topic, asymptotic representation theory. We started by studying Thoma's work [161] on the characters of the infinite symmetric group (I learned about this paper from I. M. Gelfand and I. Segal) and with the problem concerning the asymptotics of the Plancherel measure on Young diagrams, which seemed to me t o be of key importance. An important role in investigating these problems was played by our experience in ergodic theory and dynamical systems (ergodic method, invariant central measures, etc.). We worked a lot with AF-algebras and the K-theory of these algebras, and rediscovered some known facts. I think that our most important result in this field is the computation of the K-functor of the group algebra of the symmetric group [128]. Sergei's main research interest remained, however, related t o questions concerning Young diagrams, symmetric groups, and related problems of analysis and combinatorics. These interests are well reflected in the present book. Our plans included the study of asymptotic problems concerning representations of matrix groups over finite fields: this topic was outlined in the eighties, and we discussed it with A. Zelevinsky, who worked a t that time on the application of the method of Hopf algebras to a somewhat different problem about the structure of the set of representations of all finite groups GL(n, F,). However, it was not until 1996 that we started a serious study of the characters of the groups G L ( m , F,) and
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Publications of Sergei Kerov K.1. Double function algebras on a finite group, Zap. Nauchn. Sem. LOMI 39 (1974), 182-185; English transl., J . Soviet Math. 8 (1977), 136-139. K.2. Duality of finite-dimensional *-algebras, Vestnik Leningrad. Univ. 1974,no. 7 (Ser. Mat. Mekh. Astr. vyp. 2 ) , 23-29; English transl., Vestnik Leningrad Univ. Math. 7 (1979), 122130. K.3. Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux ( w i t h A . M. Vershik), Dokl. Akad. Nauk S S S R 233 (1977), 1024-1027; English transl., Soviet Math. Dokl. 18 (1977),527-531. K.4. Characters and factor representations of the i n j n i t e symmetric group ( w i t h A . M. Vershik), Dokl. Akad. Nauk S S S R 257 (1981), 1037-1040; English transl., Soviet Math. Dokl. 23 (1981), 389-392. K.5. Asymptotic theory of the characters of a symmetric group ( w i t h A . M . Vershik), Funktsional. Anal. i Prilozhen. 15 (1981), no. 4 , 15-27; English transl., Funct. Anal. Appl. 15 (1981), 246-255. K.6. Characters and factor-representations of the infinite unitary group ( w i t h A . M. Vershik), Dokl. Akad. Nauk S S S R 267 (1982),272-276; English transl., Soviet Math. Dokl. 26 (1982), 570-574. K.7. Polynomial dimension groups, Operator Theory and Function Theory, Vol. I ( B . S. Pavlov, editor), Leningrad Univ., Leningrad, 1983, pp. 185-194. (Russian) K.8. The K-functor (Grothendieck group) of the infinite symmetric group ( w i t h A . M. Vershik), Zap. Nauchn. Sem. LOMI 123 (1983), 126-151; English transl., J . Soviet Math. 28 (1985), 549-568. K.9. W-graphs of representations of symmetric groups, Zap. Nauchn. Sem. LOMI 123 (1983), 190-202; English transl., J . Soviet Math. 28 (1985), 596-605. K.lO. Experiments i n calculating the dimension of a typical representation of the symmetric group (with A . Vershik and A. Gribov), Zap. Nauchn. Sem. LOMI 123 (1983), 152-154; English transl., J . Soviet Math. 28 (1985), 568-570. K . l l . Isomorphisms of rings constructed by simplicia1 schemes ( w i t h A . Karp), Questions o f Differential Geometry "In t h e Large" ( A . L. Verner, editor), Leningrad Gos. Ped. Inst., Leningrad, 1983, pp. 60-66. (Russian) K.12. Characters, factor representations and K-functor of the infinite symmetric group ( w i t h A . M. Vershik), Operator Algebras and Group Representations, Vol. I1 (Proc. Internat. Conf., Neptun, 1980; Gr. Arsene et al., editors), Monogr. Stud. Math., vol. 18, Pitman, Boston, M A , 1984, pp. 23-32. K.13. The Robinson-Schensted-Knuth correspondence and the Littlewood-Richardson rule, Uspekhi Mat. Nauk 39 (1984), no. 2, 161-162; English transl., Russian Math. Surveys 39 (1984), no. 2, 165-166. K.14. Asymptotic behavior of the m a x i m u m and generic dimensions of irreducible representations of the symmetric group ( w i t h A . M. Vershik), Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 25-36; English transl., Funct. Anal. Appl. 19 (1985), 21-31. K.15. Locally semisimple algebras. Combinatorial theory and the KO-functor ( w i t h A. M. Vershik), Itogi Nauki i Tekhniki: Sovremennye Problemy Mat., vol. 26, V I N I T I , Moscow, 1985, pp. 3-56; English transl., J . Soviet Math. 38 (1987), 1701-1733. K.16. The characters of the infinite symmetric group and probability properties of the RobinsonSchensted-Knuth algorithm ( w i t h A . M . Vershik), SIAM J . Algebraic Discrete Methods 7 (1986), 116-124.
xiv
PUBLICATIONS O F SERGE1 KEROV
K.17. Combinatorics, the Bethe ansatz and representations of the symmetric group ( w i t h A. Kirillov and N. Reshetikhin), Zap. Nauchn. Sem. LOMI 155 (1986), 50-64; English transl., J . Soviet Math. 41 (1988), 916-924. K.18. Distribution of symmetry types of high rank tensors, Zap. Nauchn. Sem. LOMI 155 (1986), 181-186; English transl., J . Soviet Math. 41 (1988), 995-999. K.19. Random Young tableaux, Teor. Veroyatnost. i Primenen. 31 (1986), 627-628; English transl., Theory Probab. Appl. 31 (1986),553-554. K.20. Realizations of *-representations of Hecke algebras, and Young's orthogonal form, Zap. Nauchn. Sem. LOMI 161 (1987), 155-172; English transl., J . Soviet Math. 46 (1989), 2148-2158. K.21. Realzzations of representations of the Brauer semigroup, Zap. Nauchn. Sem. LOMI 164 (1987), 189-193; English transl., J. Soviet Math. 47 (1989), 2503-2507. K.22. Characters and realizations of representations of the infinite-dimenszonal Hecke algebra, and knot invariants ( w i t h A . M. Vershik), Dokl. Akad. Nauk S S S R 301 (1988), 777-780; English transl., Soviet Math. Dokl. 38 (1989), 134-137. K.23. AF-algebras of truncated Pascal triangles ( w i t h V . Volchegursky), Problems in Group Theory and Homological Algebra ( A . L. Onishchik, editor), Yaroslav. Gos. Univ., Yaroslavl, 1989, pp. 29-37. (Russian) K.24. Combinatorial examples i n the theory of AF-algebras, Zap. Nauchn. Sem. LOMI 172 (1989), 55-67; English transl., J . Soviet Math. 59 (1992), 1063-1071. Zap. K.25. A n algebra of invariants for the action of the group S p ( 2 m ) i n the algebra @M2mC, Nauchn. Sem. LOMI 172 (1989), 68-77; English transl., J. Soviet Math. 59 (1992), 10721078. K.26. Representattons, maximal with respect to dimension, of symmetric groups (with A . Pass), Zap. Nauchn. Sem. LOMI 172 (1989), 160-166; English transl., J . Soviet Math. 59 (1992), 1131-1135. K.27. T h e Grothendieck group of the infinite symmetric group and symmetric functions (with the elements of the theory of KO-functor of AF-algebras) ( w i t h A . M. Vershik),Representation o f Lie Groups and Related Topics ( A . M . Vershik and D. P. Zhelobenko, editors), Adv. Stud. Contemp. Math., vol. 7 , Gordon and Breach, New Y o r k , 1990, pp. 36-114. K.28. Random processes with common cotransition probabilities ( w i t h 0. Orevkova), Zap. Nauchn. Sem. LOMI 184 (1990), 169-181; English transl., J . Math. Sci. (New Y o r k ) 68 (1994), 516-525. K.29. Hall-Littlewood functions and orthogonal polynomials, Funktsional. Anal. i Prilozhen. 25 (1991),no. 1, 78-81; English transl., Funct. Anal. Appl. 25 (1991), 65-66. K.30. Characters of Hecke and B i m a n - W e n z l algebras, Quantum Groups (Leningrad, 1990), Lecture Notes Math., vol. 1510, Springer-Verlag, Berlin, 1992, pp. 335-340. K.31. Trzangularity of transition matrices for generalized Hall-Littlewood polynomials, Quantum Groups (Leningrad, 1990), Lecture Notes Math., vol. 1510, Springer-Verlag, Berlin, 1992, pp. 389-390. K.32. Generalized Hall-Littlewood symmetric functions and orthogonal polynomials, Representation Theory and Dynamical Systems, A d v . Soviet Math., vol. 9 , Amer. Math. Soc., Providence, RI, 1992, pp. 67-94. K.33. q-analog of the hook walk algonthm and random Young tableaux, Funktsional. Anal. i Prilozhen. 26 (1992),no. 3 , 35-45; English transl., Funct. Anal. Appl. 26 (1992), 179-187. K.34. Combinatorics of rational representations of the group G L ( n , @ )( w i t h A . N . Kirillov), Zap. Nauchn. Sem. POMI 200 (1992),83-90; English transl., J . Math. Sci. (New Y o r k ) 77 (1995), 3190-3194. K.35. Gaussian limit for the Plancherel measure of the symmetric group, C . R . Acad. Sci. Paris S6r. I Math. 316 (1993), 303-308. K.36. A q-analog of the hook walk algorithm for random Young tableaux, J . Algebraic Combin. 2 (1993), 383-396. K.37. Harmonic analysis on the infinite symmetric group ( w i t h G . I . Olshanski and A. M. Vershik), C . R . Acad. Sci. Paris S6r. I Math. 316 (1993), 773-778. K.38. Asymptotics of the separation of roots of orthogonal polynomials, Algebra i Analiz 5 (1993), no. 5 , 68-86; English transl., St. Petersburg Math. J . 5 (1994), 925-941.
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K.39. Transition probabilities of continual Young diagrams and Markov m o m e n t problem, Funktsional. Anal. i Prilozhen. 27 (1993), no. 2, 32-49; English transl., Funct. Anal. Appl. 27 (1993), 104-117. K.40. The Plancherel growth of Young diagrams and the asymptotics of interlacing sequences, Dokl. Akad. Nauk 333 (1993), 8-10; English transl., Russian Acad. Sci. Dokl. M a t h . 48 (1994);420-424. K.41. The asymptotics of interlacing sequences and the growth of continuous diagrams, Zap. Nauchn. S e m . POMI 205 (1993), 21-29; English transl.. J . Math. Sci. ( N e w Y o r k ) 80 (1996), 1760-1767. K.42. Polynomial functions o n the set of Young diagrams ( w i t h G . I . Olshanski), C . R . Acad. Sci. Paris S8r. I Math. 319 (1994), 121-126. K.43. Asymptotics of large random Young diagrams, Abstracts S i x t h C o n f . Formal Power Series and Algebraic Combinatorics, D I M A C S , 1994, pp. 285-294. K.44. Asymptotic representation theory of the symmetric group and its applications i n analysis, D. Sci. thesis, Steklov Institute o f Mathematics at S t . Petersburg, St. Petersburg, 1994; English transl., this volume. K.45. Stick breaking process generates virtual permutations with the Ewens distribution ( w i t h N . V . Tsilevich), Zap. Nauchn. S e m . POMI 223 (1995), 162-180; English transl., J . M a t h . Sci. ( N e w Y o r k ) 87 (1997), 4082-4093. K.46. Subordinators and permutation actions with quasi-invariant measure, Zap. Nauchn. S e m . POMI 223 (1995), 181-218; English transl., J. Math. Sci. ( N e w Y o r k ) 87 ( 1 9 9 7 ) ,4094-4117. K.47. Coherent allocations; and the Ewens-Pitman formula, Preprint POMI 21/1995 ( 1 9 9 5 ) , 1-15. K.48. Small cycles of big permutations, Preprint POMI 22/1995 ( 1 9 9 5 ) , 1-9. K.49. A dzflerential model for the growth of Young diagrams, T r u d y Sankt-Peterburg. Mat. Obshch. IV (1996), 165-192; English transl., Amer. Math. Soc. Transl. ( 2 ) 188 (1998), 111-130. K.50. The boundary of Young lattice and random Young tableaux, Formal Power Series and Algebraic Combinatorics ( N e w Brunswick, N J , 1994), D I M A C S Ser. Discrete M a t h . Theoret. C o m p u t . Sci., vol. 24, Amer. Math. Soc., Providence, RI, 1996, pp. 133-158. K.51. Rook placements o n Ferrers boards, and.matrix integrals, Zap. Nauchn. S e m . POhtI 240 (1997), 136-146; English transl., J . Math. Sci. ( N e w Y o r k ) 96 (1999), 3531-3536. K.52. Interlacing measures, Kirillov's Seminar o n Representation Theory, Amer. Math. Soc. Transl. ( 2 ) 181 (1998), 35-83. K.53. T h e boundary of the Young graph with Jack edge multiplicities ( w i t h A . Okounkov and G . Olshanskii), Internat. M a t h . Res. Notices 1998,173-199. K.54. O n a group of infinite-dimensional matrices over a finite field ( w i t h A . M. V e r s h i k ) , Funktsional. Anal. i Prilozhen. 32 (1998), no. 3 , 3-10; English transl., Funct. Anal. Appl. 32 (1999), 147-152. K.55. T h e algebra of conjugacy classes i n symmetric groups, and partial permutations ( w i t h V . Ivanov), Zap. Nauchn. S e m . POMI 256 (1999), 95-120; English transl., J. M a t h . Sci. ( N e w Y o r k ) 107 (2001), 4212-4230. K.56. Anisotropic Young diagrams and Jack symmetric functions, Funktsional. Anal. i Prilozhen. 34 (2000),no. 1 , 51-64; English transl., Funct. Anal. Appl. 34 (2000), 41-51. K.57. The Plancherel measure of the Young-Fibonacci graph ( w i t h A . G n e d i n ) , Math. Proc. C a m bridge Philos. Soc. 129 (2000), 433-446. K.58. The Martin boundary of the Young-Fibonacci lattice ( w i t h F. G o o d m a n ) , J. Algebraic Combin. 11 (2000), 17-48. K.59. E q u i l i b ~ u mand orthogonal polynomials, Algebra i Analiz 12 (2000),no. 6 , 224-237; English transl., S t . Petersburg Math. J . 12 (2001), 1049-1059. K.60. The Markov-Krein correspondence i n several dimensions ( w i t h N . V . Tsilevich), Zap. Nauchn. Sem. POMI 283 (2001), 98-122; English transl., t o appear i n J. Math. Sci. ( N e w York). K.61. A characterization of G E M distributions ( w i t h A.Gnedin), Combin. Probab. C o m p u t . 10 ( 2 0 0 l ) , 213-217. K.62. Fibonacci solitaire ( w i t h A . Gnedin), R a n d o m Structures Algorithms 20 ( 2 0 0 2 ) , 71-88.
CHAPTER 0
Introduction 0. Plan of the book
The first main subject of this work is the description of asymptotic properties of the characters of finite symmetric groups and the related construction of the characters of the infinite symmetric group. More generally, the combinatorial techniques developed here can be applied to a wide class of algebras that can be approximated by finite-dimensional semisimple algebras, in particular, to the group algebras of locally finite groups. The problem of computing the characters of such groups and algebras is reduced to a typical problem of potential theory on countable graphs, namely, to the description of nonnegative harmonic functions on them. Similarly to the classical situation of the unit disk, the nonnegative harmonic functions are constructed by the (generalized) Poisson integral taken over the ideal boundary of the graph. The computation of this boundary, which is similar to the classical Martin boundary, is usually a difficult problem. In this book we carry out this computation for a number of examples of combinatorial and algebraic origin related to the lattice of Young diagrams. Among all characters of the infinite symmetric group, we are most interested in the character of its regular representation. We give a detailed description of the shape of large Young diagrams typical with respect to the Plancherel measure on the finite symmetric group, and also study stochastic properties of the deviations of random diagrams from the limiting curve. In other terms, we investigate the limits of joint distributions of characters of irreducible representations of the symmetric group of degree n with respect to its Plancherel measure. The asymptotic analysis of the Plancherel measure is the second important component of this book. The third main subject is related to continuous limits of numerous nontrivial combinatorial constructions and algorithms of the theory of symmetric groups and their representations. We establish a direct relation between the transition probabilit,ies of the Plancherel measure of the infinite symmetric group and the partial fraction expansions of rational functions with interlacing zeros and poles. Using this relation, we manage to obtain a continuous analogue of the hook walk algorithm, which is well-known in the combinatorics of Young diagrams. This construction in turn provides a completely new description of the relation between the classical moment problems of Hausdorff and Markov. In probabilistic terms, the limiting algorithm (we call it the interval shrinkage algorithm) allows us to construct the distributions of integrals over random Dirichlet measures. The dynamical process of random growth of Young diagrams, which we regard as the Plancherel measure of the infinite symmetric group, leads in a natural way to a continuous model which is equivalent to the first order partial differential equation
Rt+RR,=O.
2
0.
INTRODUCTION
The typical limit shape of large random Young diagrams (we call it the arcsine law) arises unexpectedly often in seemingly unrelated problems. We give two examples of such situations: the asymptotic behaviour of interlacing roots of orthogonal polynomials and the perturbation of spectra of typical large random matrices under a linear constraint. Within these three areas (the asymptotic theory of characters of the infinite symmetric group, limiting properties of the Plancherel measure of this group, and applications of the representation theory of symmetric groups in analysis), we discuss a q-analogue of the hook walk algorithm, study in detail Macdonald symmetric polynomials, and compute the boundaries of the related deformations of the Young graph which is the central object of all our considerations. This book consists of a detailed introduction and four chapters subdivided into sections. We start with a preliminary list of the results and necessary references. A more detailed description of the contents is given below in the main part of the introduction. In the first chapter we present an account of general methods for dealing with algebras that have an approximation by finite-dimensional semisimple subalgebras (we call them locally semisimple, or, for short, LS-algebras). Conceptually, the theory of such algebras goes back to von Neumann, and the simplest examples were considered in the 60s by Glimm and Dixmier. A systematic study of these algebras was initiated by Bratteli's paper [74], which laid the foundations of the combinatorial approach t o LS-algebras and his subsequent papers [75, 76, 771. A fundamental contribution to the theory was made a little later by Elliott [88],who showed that the Grothendieck group K Oendowed with a natural ordering structure is a complete invariant of an LS-algebra. Another key fact was discovered by Effros, Handelman, and Shen [87]; they found an abstract characterization of the class of all ordered abelian groups arising as Ko(A) for some LS-algebra A. A summary of the first ten years of the theory of LS-algebras is given in the survey [86]. A new impulse for the study of LS-algebras was given by considering infinitedimensional analogues of classical groups such as the unitary group
(see [160]) and the infinite symmetric group
These examples, which are far more difficult than those studied at the initial stage of the theory, considerably influenced the development of its general techniques. The attention of mathematicians became focused on the problem of computing the characters of groups that have an approximation by finite or compact subgroups. A substantially new approach to the analysis of this problem consisted in regarding LS-algebras as crossed products associated with dynamical systems. The ergodic method for constructing characters, suggested in [13],is based on this approach, and develops A. M. Vershik's ideas applied earlier in [ l o ] . Let us also mention two other new results from the general theory of LSalgebras, which were discovered in [13] and [126] when working with the group 6,. The first one establishes the key role of the notion of semzfinite characters of LS-algebras, introduced in [13],in the description of positive elements of the
0. P L A N OF T H E BOOK
3
Grothendieck group KO. The second one consists in introducing the class of multiplicative Bratteli diagrams and developing the related theory of ordered Riesz rings. A survey of these results can be found in papers by A. M. Vershik and the author [16], [17], [128]. Continuous analogues of branching graphs and related problems are considered in [42]. The central problem of the second chapter is the approximative computation of the characters of the infinite symmetric group 6,. The characters and representations of the finite symmetric groups were found at the beginning of the twentieth century by F'robenius [61] and Scliur [155]. Necessary combinatorial techniques were developed in a long series of papers by Young; the list of these papers can be found in the last paper of the series [168]. In Weyl's famous book [8]the representations of the symmetric groups are closely intertwined with the representations of the classical matrix groups. Among the great number of textbooks and monographs treating the theory of characters and representations of B,, we would like to mention [26, 29, 50, 621, [113, 136, 1521. The monograph [81]is devoted to probabilistic problems on the group of permutations. A combinatorial basis for the representation theory of symmetric groups is provided by Young diagrams and Young tableaux. We borrow the terminology and necessary background on the algebra of symmetric polynomials from Macdonald's remarkable monograph [49]. It is worth noticing that, in spite of its hundredyear history, the combinatorics of permutations and their representations is still actively developing. We mention a combinatorial version of the Fourier transform for 6, [152], [157], [133],the hook-length formula [92] for the dimensions of its irreducible representations, various descriptions of the Littlewood-Richardson rule for the decompositions of tensor products of representations of unitary groups, the hook walk algorithm [102], [103], and a relation to the Bethe ansatz [41] (the list is not intended t o be exhaustive). The role of symmetric groups and Hecke algebras in Jones' recent construction of new topological invariants of knots and links is not less impressive, see [114], [19]. The characters of the infinite symmetric group 6 , were first found by Thoma [161], his result being surprisingly equivalent to a purely analytical problem of characterizing totally positive sequences and series which was solved independently by Schoenberg and his colleagues [64]. A completely different description of the characters of 6, obtained in [13]is a part of a wider project of Vershik for the study of approximation problems. For other realizations of Vershik's approach see, for example, his papers [20], [119],
[lo].
The problem of computing the characters of 6, is equivalent to the problem of finding nonnegative harmonic functions on the lattice (graph) of Young diagrams. In the second chapter we present a detailed study of the similar problem for a certain two-parameter deformation of the Young graph which reflects the branching of Macdonald symmetric polynomials introduced in [139], [140]. For appropriate values of the parameters of this deformation we recover the results of Kingman [129], [130] and Nazarov [51], and also obtain the Jack graphs which are used in the third chapter for computing multivariate Selberg integrals. This part of the book is based on the author's papers [34], [35],[119]. One of the first examples of describing harmonic functions on graphs is provided by de Finetti's classical theorem on random sequences invariant under finite permutations. Another important source of discrete problems on positive harmonic
4
0.
INTRODUCTION
functions is the theory of random walks on groups; see the survey by Kaimanovich and Vershik [115] for an introduction to this theory. In the third chapter we study in detail the most interesting character of the infinite symmetric group, the character of its regular representation. First we introduce the general notion of the Plancherel measure as a Markov chain on the branching graph of irreducible representations of approximating finite groups; this definition is convenient for working with arbitrary locally finite groups. Then we reproduce the old theorem of Vershik and the author [12], [la]on the asymptotics of the shape of large random Young diagrams ( a similar fact was obtained independently in [138]). Most of the rest of the results discussed later in this book are motivated by this theorem. The new results of the third chapter include the description of typical symmetries of high rank tensors [31],an analogue of the central limit theorem for the characters of the symmetric group 6, [40], and a q-analogue of the hook walk algorithm [36]. We also give a new derivation of multivariate Selberg integrals (see [156], [69], [151],[142]),based on the results of [46]. The fourth and final chapter contains the author's results related to applications of the asymptotic theory of characters of 6 , in analysis. Though they do not depend formally on the first three chapters and are of value in themselves, nevertheless the ideology and problems of the fourth chapter are motivated by the results on the Plancherel measure of 6, and the limit shape of large Young diagrams. I-t seems that there is a deep connection between the symmetric groups and the theory of functions, and our results, as well as the equivalence of the EdreiSchoenberg theorem and the theorem on the characters of G,, are manifestations of this connection. Indirect evidence of this is the work of Issai Schur, who obtained remarkable theorems in both fields. Our first result in the fourth chapter is based on the representation of pairs of interlacing sequences of real numbers by so-called "rectangular diagrams", which are generalizations of Young diagrams. This simple technique allows us to state the problem of the asymptotic behaviour of such pairs. Following [38],we prove that the limit shape of diagrams which describe the interlacing roots of classical orthogonal polynomials is the same as for large random Young diagrams. The second main result published in [37] establishes a bijection between diagrams and probability measures. Essentially the same correspondence, but obtained from completely different reasons, appears in Krein and Nudelman's book [47] in connection with the Markov moment problem, and in [82] it was used in the problem of computing the distributions of integrals over the random Dirichlet measures. The relation, discovered in [37],between the Krein correspondence and the Plancherel measure of the symmetric group 6, offers a tempting opportunity t o apply various combinatorial techniques of the theory of permutations and their representations to the moment problem and other analytical problems. Using this idea and passing appropriately to the limit in the hook walk algorithm, we obtain a new stochastic description of the Krein correspondence, the interval shrinkage process. The third important result (see [122], [40]) is the construction of a differential model of growth of Young diagrams which turns out to be equivalent to the wellstudied Burgers equation. We prove that the asymptotic behaviour established earlier for random Young diagrams holds also for this model. The relation of the
51. GENERAL THEORY O F LOCALLY SEMISIMPLE ALGEBRAS
5
differential model t o the semicircle diffusion in the free probability theory developed by Voiculescu and his pupils [166], [167], [71] is worth further studying. After these short explanations we proceed t o a more detailed exposition of our results.
$1. General theory of locally semisimple algebras The main object of our study is the infinite symmetric group 6,.
By definition,
6, is a discrete countable group which consists of finite permutations of positive integers (that is, of permutations for which almost all elements are fixed points). Denote by 6, the finite subgroup of permutations from 6, that leave the elements n + l , n+2,. . . unchanged; then the group 6, = 6, is a union of an increasing sequence of finite subgroups. Infinite groups with this property are called locally finite. Since the 70s, Bratteli, Elliott, and others have developed powerful combinatorial techniques for working with group algebras of locally finite groups. Recall the basic results of their theory. 1.1. Combinatorial theory of LS-algebras. A locally semisimple algebra (LS-algebra for short) is the inductive limit
of an inductive family
of finite-dimensional semisimple algebras U,. We will assume that all homomorphisms preserve unity. According to Wedderburn's theorem, each finite-dimensional semisimple algebra is isomorphic to a direct sum of full matrix algebras:
Here I?, is a finite set whose elements index the equivalence classes of complex irreducible representations nx of the algebra Q,, and d(X) = dim nx is the dimension of the irreducible representation corresponding to X E I?,. A finite-dimensional seniisimple algebra 8, is determined up t o isomorphism by the list of dimensions {d(X))xEr, of its irreducible representations. An embedding of finite-dimensional semisimple algebras
can be adequately described by a bipartite graph with vertex set Tn-1 U I?,. By and A E I?, are joined by an edge of multiplicity definition, the vertices X E x(X,A), where x(X,A) is the multiplicity of xx in the decomposition
We denote by n~ the irreducible representation of the algebra U, corresponding to A E .,?I By F'robenius reciprocity, the same multiplicities arise in the decomposition
0.
6
INTRODUCTION
of the induced representation
If the bipartite graphs of two homomorphisms i n ,j,: + 2, coincide, then there exists an invertible element b E U, such that j,(a) = bi,(a) b-' for all a E 24-1. It follows that an inductive family (1.1) of finite-dimensional semisimple algebras generates an infinite graph, called the branching graph (= Bratteli diagram) of this family. By definition, I' is the union of the bipartite graphs associated with the embeddings in: UnP1 + U,. The vertex set of I' is I' = UTz0r n .
FIGURE 1. The Young graph Our most important example is related t o the inductive family
of the group algebras of the finite symmetric groups 6,. For convenience, we introduce the group algebra @[eO] -. @ of the nonexisting group Go; we identify it with The branching graph of this family is the Young graph shown in the algebra @[el]. Figure 1. Its vertices are Young diagrams, and the edges connect pairs of diagrams which differ from each other by a single square*) . The Young graph describes the branching of irreducible representations of the symmetric groups under restricting *) We write X
/* A if a Young diagram A is obtained from a diagram X by adding one square.
6 1 . GENERAL THEORY O F LOCALLY SEMISIMPLE ALGEBRAS
7
to a subgroup, and inducing
Note that all edges of the Young graph are simple. 1.2. Characters of LS-algebras and harmonic functions on graphs. The branching graph contains exhaustive information on an inductive family of finite-dimensional semisimple algebras, and hence on the LS-algebra U, = 1 4 '21, which is the inductive limit of this family. In particular, the characters of the algebra U, can be described in terms of the branching graph. A character of an LS-algebra U, is a linear functional
that enjoys the following three properties: (a) (positive definiteness) yi(aa*) 0 for all a E a,*) ; (b) (centrality) +(ab) = yi(ba) for all a , b E U,; (c) (nornlalization) $(I) = 1. In particular, the restriction of a character of the algebra U, gebra Un can be written as
>
+
to the subal-
Comparing with the decomposition
we conclude that the coefficients p(X) satisfy the condition
for all X E r,-l. DEFINITION. Let y:I? 4 @ be a function on the vertex set of a branching graph I?. We call it harmonic if (2.2) holds for all n = 1 , 2 , . . . . In our considerations, fornzula (2.2) plays the role of the mean value theorem for ordinary harmonic functions. The following simple result establishes a bijection between the characters of an LS-algebra and the harmonic functions on its branching graph. *) Here a H a* stands for a n arbitrary involution in Q, which is compatible with the standard involutions on the matrix subalgebras U ' , n = 1 , 2 , . . . . Although such a n involution is not unique, condition (a) does not depend on its choice.
8
0.
INTRODUCTION
THEOREM 1 ([17],59, Theorem 5). For each nonnegative harmonic function cp: r + R+ normalized so that 4 8 ) = 1 *), there exists a unique character $: U ', + C of the corresponding LS-algebra such that
for all n = 0 , 1 , 2 , . . . . Fornula (2.1) establishes a bijective homeomorphism between the space of characters of the LS-algebra Char('U,) and the space Harm(r) of normalized nonnegative harmonic functions on its branching graph. 1.3. The boundary of a branching graph. The space of harmonic functions Harm(r) endowed with the pointwise convergence topology is convex, compact, and metrizable. By Choquet's theorem (see, for example, [59]),each function cp E Harm(F) can be represented as
where M is a probability measure supported by the set of extreme points &(I?)= ex Harm(r). In fact, decomposition (3.1) with this property is unique; that is, the convex compact set Harm(r) is a simplex. This follows, for example, from Effros' results ( [ 8 6 ] ,Lemma 4.3). In all examples considered in this paper the set of extreme points &(I?)is closed in the simplex Harm(r). We call it the boundary of the branching graph r . Formula (3.1) allows us to identify the simplex of nonnegative harmonic functions Harm(r) with the set of all Bore1 probability measures on the boundary &(I?)of the branching graph. According to a classical theorem (see, e.g., [24], Chapter 1, Theorem 3.5), a nonnegative harmonic function on the disk has a unique Poisson integral representation
and the representing measure p can be obtained as the radial limit dQ p(d0) = lim cp(reie) r-1 2~
In Chapter 1 we establish similar facts for branching graphs. Towards this end, we define a generalized Poisson kernel @(X,6),where X E r, 6 E E ( r ) , and the radial embedding i: r --, E(I'), which possess the following properties: (a) For a fixed 6 E E ( r ) , the function cp(.) = a(.,6) lies in the simplex Harm(r). (b) The functions cPx(.) = @(A, .) are continuous on the boundary E ( r ) and distinguish the points of E ( r ) . (c) Given 6 E E, denote by Mn the probability distribution on the nth level rn of the branching graph I? with weights
*)We denote by 0 the unique point of the set
ro.
s l . GENERAL T H E O R Y O F LOCALLY SEMISIMPLE ALGEBRAS
9
where d(X) stands for the rl~imberof directed paths corlnecting Q) and A. Then the distributions i(hl,) weakly converge in & to the unit weight at tlie point 6 E &. Under these assumptions the following theorem holds. THEOREM 2 [123]. Conditions (a)-(c) i m p l y ( I ) each harmonic function cp E Harm(r) has a unique Poisson integral representation
uih,ere /L is a Borel probability measure o n th,e boundary &(I?); and (2) the representing measure p i s the weak limit of the discrete probability distributions i(ll.fn), where the measure M,,, i s determined by the h a r m o n i c function cp by form,ula (3.4). 1.4. C e n t r a l measures. Historically, one of the first problems reducible to compliting harmonic fiinctions on a graph was the problem of describing all Borel probability measures, on the space of 0- 1 sequences, that are invariant under finite permutatiorls of coordinates. The prod~ictmeasures &Ip, with a common probability p E [0, 11 of the unity for all coordinates, are examples of such measures. According to de Finetti's theorem (see [58],Vol. 2, Chapter 7, §4), each invariant measure M is a mixture of the product measlires:
where p is a urliqliely determined probability distriblition on the unit interval. Let us identify 0 - 1 sequences with paths in the Pascal triangle (see Figure 2). Then the invariance of M is eqliivalerlt to the following condition: ( C ) For any two paths u, I J of the graph I? from the initial vertex 0 to a common vertex X E I?, the hl-probabilities of u and v coincide and depend only on A. DEFINITION. A Borel probability measure M on the path space of a branching graph I? is said to be central if is satisfies condition (C). Denote by cp(X) the common probability (from corldition (C)) of all paths ending at X E I?. It is easy to check that cp is a harmonic function. This correspondence establishes a bijection between the space Cent(r) of central measures and Harm(r). Let 11sdenote by d ( u , A) the number of directed paths in I? from u E I? to X E r, and call it the dim,ension of the interval [u, A]. If u = 8 is the initial vertex of the graph, then we write simply d(X) = d(0, A). Given a fixed positive integer n, associate with a harmonic furlctiorl cp E Harm(r) the probability distribution on r,,,deiined by
The family of distributions {Mn),"==, is coherent in the following sense:
0.
INTRODUCTION
FIGURE 2. The Pascal triangle. Coherent families provide a convenient language for dealing with central measures. In particular, the mixing measure p in the Poisson integral (3.5) arises as the weak limit of Mn. In a special case, coherent families were introduced by Kingman [129] under the name of partition structures. Each central measure is a Markov measure; it is determined by transition probabilities p ( X , A), X / A, defined on the edges of the graph. 1.5. Ergodic method. A harmonic function cp E Harm(r) determines an extreme point of this simplex if and only if the corresponding central measure M is ergodic with respect t o the tail equivalence relation J in the path space T of the graph I?. By definition, the equivalence classes of J consist of eventually coinciding paths. The ergodicity of M means that, for each measurable set A C T consisting of entire equivalence classes of J, its measure M(A) is either 0 or 1. In general there is no natural transformation S : T 4 T whose orbit partition would coincide with the tail equivalence relation I. However, one can restate Birkhoff's pointwise ergodic theorem so as to make the choice of such a transformat ion unnecessary. To this end, in the ordinary statement of the theorem, replace the Cesaro mean f ( S k t ) of a function f : T 4IR by fn(t) =
where the sum is over all paths s E T that coincide with t starting from the nth level r n . A version of the standard argument (see, e.g., [46], Addendum 3) leads to the following important result.
THEOREM 3 [13]. Let M be an ergodic central measure in the path space T of , . . , un, . . . ) such that a branching graph I?. Then the set of paths t = ( u ~u2,.
$1. GENERAL THEORY O F LOCALLY SEMISIMPLE ALGEBRAS
11
(a) there exist the limits
(b) the function cpt: I? is of full M-measure.
~(L,v) lim n+md(O, vn) '
r;
cpt(X)
=
+ E%
is harmonic and related to the measure M by (3.4)
The ergodic method for constructing the boundary of a graph is based on this theorem.
For each extreme harmonic function cp E l(r)there exists a path COROLLARY. t E T such that cp coincides with the function cpt defined by (5.2). It is worthwhile to restate this result in terms of characters.
Ur=l +
COROLLARY [13]. Let E exChar(G,) be an extreme character of a locally finite group G, = G,. Then for each n = 1 , 2 , .. . there exists a character $, of an irreducible representation of the group Gn such that
Let us call a path t E T regular if the limits (5.2) exist. The corresponding limiting function cpt is harmonic, though not necessarily extreme. According to the above corollaries, the computation of extreme harmonic functions and characters can be derived from the description of regular paths of the branching graph.
1.6. Multiplicative branching graphs. Many interesting examples of branching graphs can be obtained by using commutative graded rings; we call these graphs multiplicative. For instance, consider the graded ring R = Rn of symmetric polynomials in infinitely many variables. The Schur functions sx(x), X E yn, form a Z-basis in the additive subgroup R, of polynomials of degree n. By Pieri's well-known formula,
@r==o
and we arrive at another description of the Young graph y . If {fx)xEy is an arbitrary graded basis in R such that the coefficients of all expansions
are nonnegative, we can define a branching graph with vertex set y by declaring that a pair of vertices (A, A) is connected by an edge of multiplicity x(X, A) provided that x(X,A) # 0. We emphasize that t o state the problem of computing harmonic functions on a graph, there is no need to require that the multiplicities of edges be integers. In Chapter 2 we study the multiplicative branching graphs which form a twoparameter deformation of the Young graph y . In this case the basis { f x ) consists of the Hall-Littlewood-Macdonald symmetric polynomials Px(x; q , t ) introduced in [139, 1401.
12
0.
INTRODUCTION
General multiplicative branching graphs satisfy the following extremality criterion. THEOREM4 [126]. Given a harmonic function cp E Harm(r) on a multiplicative branching graph I?, let $: R + R be its extension (by linearity) to the ring R (i.e., $(fx) = p(X) for X E r). Then the following conditions are equivalent: (a) cp E &(I?)is an extreme point of the simplex Harm(r); (b) $: R + IR is a ring homomorphism. Special cases of this criterion were used by Thoma [I611 and Voiculescu [165]. COROLLARY. The boundary &(I?) of any multiplicative branching graph closed in the simplex Harm(r).
r
is
Since all branching graphs considered in this book are multiplicative, the boundary &(I?)is well-defined as a compact topological space, and the harmonic functions are in a one-to-one correspondence with Bore1 probability measures on the boundary.
1.7. The crossed product construction. In Section 1.1 above we have associated a branching graph with an arbitrary family
of finite-dimensional semisimple algebras over @. Conversely, assume that a graded I?, with directed edges enjoys the following properties: graph = (a) each edge (A, A) beginning at the nth level X E r, leads to the next level, i.e., A E (b) 0 E rois the only vertex with no ingoing edges; (c) each vertex has a t least one outgoing edge; and (d) all levels r,, n = 1 , 2 , . . . , are finite. Then it is not difficult to construct an inductive family (7.1) with branching graph r. By definition, the algebra U, in this family consists of matrices of the form
Ur=l
(7.2)
a
=
(a,,),
u , v E Tn(X),
E
r,,
where T,(X) stands for the number of paths of length n ending at X E r,. Define the loop space B = B ( r ) of the branching graph r as the graph of the tail equivalence relation 5 in the path space T. Thus B consists of pairs of eventually coinciding paths s, t E T; a sequence (s,, t,) E B converges to (s, t) if lims, = s, limt, = t in T, and there exists a level rN such that s, and t, coincide for all n N. The loop space B is locally compact and totally disconnected. The limiting LS-algebra U, = 142, can be identified with the algebra of compactly supported locally constant functions a: B + @ on the loop space. The multiplication is defined by
>
where r runs over all paths that are 5-equivalent to s , t. There is a natural involution a*(s,t)
= a(t, s).
This construction of an LS-algebra is a version of von Neuinann's construction of the crossed product associated with a dynamical system (cf. [150]). In particular,
51. GENERAL THEORY O F LOCALLY SEMISIMPLE ALGEBRAS
13
it plays an important role in constructing realizations of factor representations of LS-algebras. For the infinite symmetric group, such realizations were found in [14, 151. Note also that diagonal "matrices" a E % , form a maximal commutative subalgebra of this algebra, and that all characters are states of measure type with respect to this subalgebra:
where p is a central measure on the path space T. 1.8. Dimension groups. A branching graph is associated with an inductive family of algebras (7.1) and depends not only on the limiting LS-algebra a ,, but also on the choice of the approximating family. Though LS-algebras are quite varied, and the problem of their classification is not very interesting, the limiting algebra % , has a nontrivial invariant which reduces all problems concerning LS-algebras to problems related to abelian groups. This invariant was suggested by Elliott [88]; it is the Grothendieck group KO(%,) endowed with a natural ordering structure. By definition, the genera, and tors of this group are the classes [p] of unitarily equivalent projections p E ,% the relations are of the form
where p,q are orthogonal projections. Note that this description uses a specific property of the K-functor on the class of LS-algebras: with the standard definition, the group KO(%,) is generated already by the classes of submodules of the free module with one generator. The semigroup K$(%,) generated by the classes [p] is a cone and defines an ordering structure in the group KO(%,). The class of the unity projection [l]E K$(%,) determines an order unity in the cone K$(%,), i.e., it is contained in no proper order ideal of the Grothendieck group. Elliott obtained the following important result. THEOREM5 [88]. The triple (KO(%,), K$(u,), [I]), which is an ordered abelian group with a distinguished order unity, is a complete invariant of the LSalgebra .Q , Elliott's invariant is continuous with respect t o inductive limits: if % ,
l&
=
an,then
Consequently, it can be easily described in terms of the branching graph I? associated with the algebra 8,. A dimension function on r is any Z-valued function which is defined for almost all vertices of the graph and satisfies the equation
on its domain. We identify two functions f l , fi if they differ only on a finite set of vertices. A dimension function is called virtually positive if it is nonnegative for
14
0.
INTRODUCTION
almost all vertices of I?. A distinguished example of a virtually positive function is the function d(X), X 6 I?, defined in Section 1.4 above. It is easy to see that the group KO(%,) is identified with the group of (classes of) dimension functions; the cone K:(%,), with the cone of virtually positive elements; and the order unity [I], with the function d: r -+N*) . It was a remarkable discovery that Elliott's invariant can be characterized in intrinsic terms. Recall (see [22, 941) that an ordered abelian group G is called a Riesz group if it satisfies the following interpolation axiom: (I) If four elements of the group G are related by the inequalities
fi
forall
i,j=1,2,
then there is an interpolating element h 6 G such that
fi
forall
i,j=1,2.
Effros, Handelman, and Shen showed [87] that the dimension groups of LSalgebras can be essentially characterized as Riesz groups.
THEOREM 6 [87]. The class of the dimension.groups of LS-algebras coincides with the class of Riesz groups that satisfy the following property of purity: if g E G and ng 6 G+ for some n > 0, then g E G+ too. This theorem allowed several authors to obtain a number of interesting counterexamples in the theory of C*-algebras by presenting appropriate Riesz groups (see [72, 781). An explicit construction of the corresponding algebras seems to be rather difficult. A dual characterization for the cone G+ of positive elements of a Riesz group G in terms of positive functionals on this group was obtained by Vershik and the author ([17], Theorem 21). The same survey [17] develops the theory of Riesz rings and modules and presents a number of examples of descriptions of dimension groups. The computation of the dimension group for the infinite symmetric group 6, was first carried out in [126]. It is worth mentioning that Poincark's paper [147] can be regarded as an example of computing a dimension group. Note that the Riesz group technique, which was originally developed for describing the K-functor of locally semisimple algebras, is now being efficiently applied to the classification of automorphisms in symbolic dynamics (see [132], as well as outstanding recent results of Skau, Giordano, and Putnum [96]). 1.9. Semifinite traces and other invariants of LS-algebras. We have described two general methods for dealing with algebras that can be approximated by semisimple matrix subalgebras. The first one is based on studying the combinatorial properties of the branching graph associated with the approximation; and the second one, on using ordered Riesz groups. In principle, both techniques can be applied to any problem concerning LS-algebras. The central problem of our work is the description of finite characters of LSalgebras. It is closely related to the problem of describing semifinite characters which can be understood as the characters of two-sided ideals of LS-algebras. Semifinite characters play a key role in the characterization of the cone of positive elements of the dimension group of an LS-algebra (see [17], Chapter 2, $4). *) In connection with this description, the ordered group KO(%,) is called the dimension group.
51. GENERAL THEORY O F LOCALLY SEMISIMPLE ALGEBRAS
15
The problem of describing semifinite characters was first posed in [126];an exhaustive description of semifinite characters for the infinite symmetric group 6, was obtained in [13].
DEFINITION.A semifinite character of a locally semisimple algebra 2, --t [0, m ] , defined on the cone
is a
function X : U&
(9.1)
UL
=
{aa*: a E U,),
satisfying the following four axioms: 1. if a , b E U L , then ~ ( ab) = ~ ( a ) ~ ( b ) ; 2. if a E 82 and a E R+, then ~ ( a a=) a x ( a ) ; 3. if a E UL, then ~ ( a a *=) x ( a * a ) ; 4. for each element a E U L l the value ~ ( a is) the least upper bound of the numbers ~ ( bover ) all b a with ~ ( b<) m .
+
+
<
The set I ( x ) of elements a E U& with ~ ( a a *<) m is a self-conjugate two-sided ideal in U,, called the ideal of finiteness of the semifinite character. Denote by I 0 ( x ) the kernel of X , i.e., the self-conjugate two-sided ideal consisting of elements a E I ( x ) with ~ ( a a *=) 0. Let U, be the crossed product associated with the branching graph r as described in Section 1.7. The algebra U, has a unique maximal commutative subalgebra which consists of diagonal quasimatrices a ( s , t ) = 6,,t a ( s ) . Semifinite characters, like finite ones, are states of measure type with respect to this subalgebra. This means that they can be written in the form
where p is a semifinite central measure on the path space T. DEFINITION. A central measure p on the path space T of a branching graph l? is called semifinite if for every cylinder set A c T there exists a covering A = Uz==l A, of A with countably many cylinder subsets A, of finite p-measure. In terms of potential theory, a semifinite character is determined by a nonnegative harmonic function cp: r' --t [0, m ] . The set I(cp) of vertices X E r with c p ( X ) < m is an order ideal of the partially ordered set r*),called the ideal of finiteness of the function cp. Each finite nonnegative harmonic function cp defined on an order ideal I c r of a branching graph r has a canonical extension to a finite or semifinite harmonic function @ on the whole graph I?. This extension can be obtained by formula (47) from [128]:
where the sum is over all edges X /" A of r such that A E I, X $ I . Semifinite characters are of interest as the traces of factor representations of von Neumann type 11,, as well as in connection with the following characterization of the cone in the Grothendieck group K O[U,]. *)The vertices of the branching graph r are ordered in an obvious way (so that 0 E ro is a minimal element). A subset I C I? is called an order ideal if X E I and X /* A imply A E I.
16
0.
INTRODUCTION
THEOREM 7 (Positivity Theorem) [126]. A dimension function f E KO[%,] is virtually positive i f and only if there exists an order ideal which contains f and such that $ ( f ) > 0 for every semifinite positive functional $:I KO[%,] -t E% which is finite and nonvanishing on this ideal. We conclude this section by mentioning a number of aspects of the theory of LS-algebras which are not considered in this book. The problem of classification of two-sided ideals in an LS-algebra is comparatively simple. Dooley's results [83]give a description of these ideals in terms of the branching graph; in [17], Chapter 2, 53, the ideals are studied in terms of Riesz groups. The cohomology groups of the tail equivalence relation of a branching graph are described in [17], Chapter 1, §12. The problem of realization of factor representations of LS-algebras is considered in [160], [14, 15, 191. Another approach to the representation theory of LSalgebras related to the classical infinite-dimensional groups and symmetric spaces was developed by Olshansky [53]. We now proceed to apply the general methods presented in this section to a number of important examples; first of all, to the infinite symmetric group. $2. C h a r a c t e r s of G,,
a n d t h e Young g r a p h
2.1. T h o m a ' s results. Elmar Thoma [161] was the first t o pose the problem of classification of characters of the infinite symmetric group 6, and t o obtain a complete solution of this problem. The novelty of the situation was due to the fact that this group is not tame: it has factor representations of von Neumann type 111 and possesses a typical collection of properties which are "wild" from the viewpoint of the traditional representation theory, see [43], 58.4. In another paper [162], Thoma showed that most discrete groups are wild; the only exceptions are almost commutative groups, i.e., groups which have a commutative subgroup of finite index. For a wild group, the classification of irreducible representations is not a reasonable problem. The central objects in this case are representations of finite von Neumann type. These representations have everywhere defined traces and can be adequately classified in terms of their characters. Thoma was the first to demonstrate this, by obtaining such a classification for the infinite symmetric group. Recall that a character of a discrete group G is any function X: G -+ @ enjoying the following three properties: (a) n
for any elements g l , . . . ,g, E G and any complex numbers 21,. . . ,z, E @ (positive definiteness); (b) ~ ( g h = ) ~ ( h g for ) all g, h E G (centrality); (c) ~ ( e = ) 1, where e is the unity of G (normalization). Denote by Char(G) the space of characters of the group G endowed with the pointwise convergence topology. This is a convex compact set; hence each character x E Char(G) is the barycentre of a certain measure p supported by the set of
52. CHARACTERS OF Em, AND T H E YOUNG GRAPH
extreme points &(G)= ex Char(G):
As we have already noted in $1, the decomposition with this property is unique; thus the space Char(G) is a simplex. Just as in the case of finite groups the set &(G) consists of the normalized characters of irreducible representations, in the case of infinite discrete groups it consists of the normalized traces of factor representations of finite von Neumann type. One of the most important parts of the problem of computing cliaracters is to find a parametrization of the set &(G) of extreme characters of tlie group G. Thoma's answer for the infinite symmetric group G = G, is the following.
THEOREM 8 [161]. Consider the space of pairs of sequences
with the componentwise convergence topology. Then the set of extreme characters E(6,) is homeomorphic to A. The character x,,p E & ( e m ) corresponding to a pair ( a , p) E A is determined by its values
on one-cycle permutations*) by the formula (1.4)
xa,0(wnt1,
x ~ ; P ( w )= Z
where w = w,,w,, permutations.
. . . is the factorization o f w
E
6, into the product of one-cycle
2.2. Totally positive sequences and series. The first step of Thoma's proof consists in reducing the original problem of computing characters to a purely analytical problem of describing totally positive series. Then Thoma solves the latter problem**) using Nevanlinna's theorem on the distribution of values of meromorphic functions. The problem of totally positive series consists in describing totally positive infinite Toeplitz matrices.
DEFINITION.A matrix (hij)i,j20 is totally positive if all its finite order minors are nonnegative. Totally positive matrices and kernels were studied, in particular, in the monographs [116], [23]. They arise in various problems of analysis and probability theory, for example, as the inverse matrices of normal Jacobi matrices, in the theory of Chebyshev systems, as the transition densities of one-dimensional Markov processes. We will be interested in Toeplitz matrices (hj-i)i,,>o - associated with one-sided sequences {hn)r=o padded with zero values for negative indices: h, = 0 for n < 0. *)Here w, E 6, is an arbitrary permutation with a unique nontrivial cycle of length n. **)Not knowing that the solution had been published 12 years earlier by Edrei [84].
18
0.
A sequence {hn},",o matrix
INTRODUCTION
is called totally positive (or a P6lya frequency sequence) if the
is totally positive. One can show that this condition follows from a formally weaker requirement that the minors of the form
i.e., corresponding t o arbitrary rows and consecutive columns, be nonnegative. Finally, a formal series H ( z ) = Cr=P=o h, zn is totally positive if the sequence of its coefficients {h,}r=, is totally positive. A remarkable description of totally positive series was suggested by Schoenberg and proved by Edrei [84].
THEOREM 9. The class of totally positive formal series normalized so that H ( 0 ) = 1 coincides with the class of the Taylor series of meromorphic functions of the form
>
>
where y 0, a, 0, pi 2 0, and x(ai+ P i ) a nonzero radius of convergence.
< m. I n particular, these series have
The most difficult problem here is t o prove that H ( z ) = eYZ provided that a totally positive series H ( z ) defines an integral function without zeros. This fact can be derived from the following result of Nevanlinna's. Given a n integral function f , denote by zia),z p ) , . . . the sequence of all points such that f (z!")) = a . By definition, the index of convergence of the value a is the number
<
THEOREM 10 11441. If pa p arcd pb 5 p for two distinct va1u.e~a integral function f , then the order o f f is at most p.
#b
of a n
Karlin ([116], p. 453) observed that it is undesirable to use such a deep, but not elucidating, result in the proof of the Schoenberg-Edrei theorem. In Section 3.3 below we present a completely different proof which is based on the approximative theory of characters of the group 6,. The idea of Thoma's reduction is based on the multiplicativity of extreme characters. It turns out that if x f(6,) and w1, w2, . . . , w, are permutations with disjoint (nontrivial) cycles, then (2.5)
x ( w 1 . . . w,) = x(w1). . . x ( w n ) .
$ 2 . C H A R A C T E R S OF
8,. AND T H E YOIJNG G R A P H
Thus a n extreme character of the group 6, values
19
is uniquely determined by its
on one-cycle permutations. It is convenient t o use the formal generating series S ( z ) = Cr=o=, p,,zn / n and its formal exponential H ( z ) = eS(").
(2.7)
The coefficients of the series H ( z ) = ~ for instance,
~ hnzn x and o S ( z ) are polynonlially related;
where thc slim is over all partitions p = (lrl2r2 . . . ) of the number n. Thoma proves that the series S ( z ) determines ail extreme character x E E(6,) if and only if thc series H ( t ) is totally positive.
2.3. Approximative description of the characters of 6,. For the Thoma used the zeros and poles of parametrization of a character x E &(em), the power series
where UJ, = ( 1 , 2 , . . . , n) E 6, is a permutation with a single nontrivial cycle of length n. Vershik and the author [13] found a more adequate interpretation of the Thoma parameters which is related to the approximation of extreme characters of the group 6, by the characters of irreducible representations of the finite symmetric slibgroups 6 , . Recall that the irreducible representations of the group 6, and their characters have a standard parametrization by partitions of the number n. It is conX2 . . . XI, by the Young diagram venient to represent a partition n = XI X = (XI, X2,. . . , X k ) containing Xi squares in the i t h row, i = 1 , 2 , . . . , k . Denote by yn the set of Young diagrams with n squares; by TA the irreducible representation of 6, corresponding t o X E y,; and by d(X) and X A the dimension and the character of this representation, respectively. Note that the notation d(X) agrees with that of Section 2.4, which was introduced from combinatorial considerations. We denote by A' the conjugate of A, i.e., the Yoling diagram whose rows coincide with the col~imnsof A. By the second corollary in Section 1.5, each extreme character x E E(6,) has a pointwise approximation by the characters of irreducible representations of the slibgrolips 6,. A detailed descriptioii of the conditions of this approximation is as follows.
+ +
+
THEOREM 11 (Approximation Theorem) [13]. Consider a sequence of growing Young diagrams A(") E y,, n = 1 , 2 , . . . . The following conditions are equizjalent: (1) The 1in1,its liin
n-bm
XX(") (w)
d(X(,))
=
~(111)
0.
20
INTRODUCTION
exist for all permutations w E 6,. ( 2 ) The limits of the relative row and column lengths
Xp)
lim - = a k , n
n-oo
lim ----- n
n-oo
exist for all k = 1 , 2 , . . . . The limiting character i n (3.2) coincides with the character xa,g i n Thoma's theorem. (Section 2.1), the elements of the sequences a , p being exactly the limits (3.3). The limits a , /? from (3.3) are called the frequencies of the character x,,p E E(6,). The frequencies provide a natural interpretation of the Thoma parameters which is completely different from their analytical meaning (zeros and poles of a generating series). Let us give an equivalent description of harmonic functions on the Young graph y . The extended Schur functions*) play here the role of the Poisson kernel from Section 1.3, and the embedding
determines the "radial projection" of the graph
y onto its boundary A.
THEOREM 12 [123]. Each harmonic function cp graph Y can be uniquely represented i n the f o m (3.5)
P(A) =
/
n
s h ( a ;a ) d p ( a ;13).
E
Harm(y) on the Youn,g
E Y,
where s x ( a ;/3) is the extended Schur function, and p is a Bore1 probability measure on the space of frequencies A. The distribution p is the weak limit of the measures i(Mn), where Mn(X) = d(X) . p(X) for X E y,. Since the problem of describing the characters of 6 , and the problem of describing totally positive series are equivalent (at the level of statements), the proof of the approximation theorem provides a new proof of the Schoenberg-Edrei theorem which does not use Nevanlinna's theory. Recently a young Moscow mathematician, Andrei Okounkov, suggested [54] a third original proof of Thoma's theorem. The interrelations of all three proofs are worth further investigating. Let us sketch the key ideas of our proof of the approximation theorem. 2.4. Asymptotics of the number of skew Young tableaux. A path in the Young graph from the empty diagram 0 E Fo to a Young diagram X is called a Young tableau of shape A. A path that connects two arbitrary diagrams X C u is called a skew tableau of shape u \ A. With the general notation of Section 1.4, the number of skew tableaux is equal to d(X, u). Following the ergodic method from Section 1.5, we must find the limits of the form
lim n-oc,
d(A, u ( ~ ) ) = cp(A), d(0,d n ) )
A
E
Y.
*)See Section 2.4 below for the definition of extended symmetric functions.
$2. CHARACTERS OF G,,
AND T H E YOUNG GRAPH
21
In Chapter 2 we will obtain exact formulas for d(X, v) / d(8, v) which are convenieiit for passing to the limit. In order to describe these formulas, let us use the modified Frobenius coordinates (fl,. . . , fd; 91,. . . ,gd) of the Young diagram v. Their definition should be clear from Figure 3 (see a precise definition in the main text). We will also need to extend symmetric functions to a wider domain. This can be most easily done for the Newton power sums
The extended power sums are the following polynomials in two sets of variables a , /3 and a separate variable y:
and
FIGURE3. The Frobenius parameters. Other symmetric functions can be uniquely represented as polyiiomials in the power sums; hence their extended versions are also defined. For example, the extended Schur ,functions can be defined by Frobenius' formula:
2 T 2 ,. . ). See [128] for equivalent, but purely combinatorial, dewhere p = (lT1, scriptions of the extended Schur functions. We usually assume that y = 1C(ai4-Pi); in this case we write simply p, (a;P), s x ( a ;P), etc.
22
0.
INTRODUCTION
The extended versions of the complete homogeneous symmetric functions are defined by the generating series
We may compute the values of symmetric polynomials on Young diagrams by applying the extended versions of these polynomials to the Frobenius parameters. For example, (4.4)
def SX(V)
=
sx(f1,. . .
, f d ; 91,. . . ,gd)r
where ( fl , . . . , f d ; g1, . . . ,gd) are the Frobenius parameters of a Young diagram v. The purpose of these definitions is the following lemma.
LEMMA.For each Young diagram X E yn there exists a symmetric polynomial Qx of degree deg Qx < n such that
where N = v l is the number of squares of v E
y~
Example:
For ordinary Young diagrams, the denominator d(0, u) can be computed by the remarkable hooklength formula (formula (6.12) in Section 3.6). Together with the above lemma, this formula gives a convenient expression for the number of skew Young tableaux d(X, u). This simple combinatorial fact seems to be new. The proof of the approximation theorem follows easily from the lemma.
2.5. Semifinite characters of 6,. The group algebra C[6,] of the infinite symmetric group is locally semisimple and has a canonical involution
This algebra is uniformly dense in the group C*-algebra C*(6,).
A description of the space of closed two-sided ideals in the group C*-algebra C*(6,) is obtained in [13]on the basis of the general theory of LS-algebras. Here we will give an equivalent description of the order ideals of the Young graph. Recall that a Young diagram is a finite subset X c N2 such that each "square" (i,j) E X is contained in X together with all squares (k, 1) with k 5 i, 1 j . Denote by y, the set of infinite diagrams, i.e., of infinite subsets I C N2 with the same property. Figure 4 shows an example of an infinite diagram. Let us endow Y , with a qiiasicompact (non-Hausdorff) topology by taking the sets
<
52. CHARACTERS O F Boo,AND T H E YOUNG GRAPH
FIGURE4. An infinite Young diagram. as the basis of rieighbourhoods.
THEOREM 13 [13]. The primitive order ideals of the Young graph are of the form
The primitive spectrum *) Prim C*[G,] to the space y,.
of the group C*-algebra is homeomorphic
We now proceed to the description of semifinite characters of the infinite symmetric group. According to the general theory presented in Section 1.9, it suffices to find extreme semifinite harmonic functions on the Young graph. A semifinite character x of a discrete group G is any semifinite trace on the group C*-algebra C*[G,]. For locally finite groups the conventional definition (see [27],6.1.1) reduces t o that of Section 1.9. Consider a proper infinite diagram I E y,, I # N2 (Figure 4), and let k (respectively, 1 ) be the number of its infinite rows (respectively, columns). Denote by u the part of I that is not contained in the union I. of its infinite rows and columns. Pick two sequences of nonnegative numbers (5.4) with
a = (a1 2 a;?2
. . . 2 a k ) and p = (PI2 p2 2 . . . 2 pl)
C(ai+ pi)= 1, and define a function pi,p:y
+
[0, CO] by setting
where sx\,(a; P) is the extended Schur function. THEOREM14 [13]. Formula (5.5) provides a general (up to a po~itivefactor) extreme semifinite harmonic function on the Young graph. It is indexed b y a pair a , p of finite sequences of nonnegative numbers with unit sum and a nonempty Young diagram u E y. *)See [27], 3.1.5, for the definition.
24
0.
INTRODUCTION
2.6. Virtually positive functions on the Young graph. Consider an arbitrary function fm: ym + R, and extend it to Young diagrams with a larger number of squares by the recurrence relation
The vector f m is said to be virtually positive if the functions f, are nonnegative for sufficiently large n (f, can be virtually positive even if it is not positive itself). The problem of describing virtually positive functions is equivalent to that of describing the cone in the K-functor Ko(C*(6,)) of the infinite symmetric group; this description was obtained in [126]. In order to state the answer, we define in the usual way (see [95])the differential operators E,, v E y, in the ring R of symmetric functions in infinitely many variables:
Here we use the standard scalar product in R with respect to which the Schur functions s~ are orthonormal. Given a vector fm(X), X E Ym, construct a symmetric polynomial P, E R by setting
; v Ey. The jet of the polynomial P is the collection of its "derivatives" ( E w P ) ( aP), Given a pair of positive integers k,1 0 (not vanishing simultaneously) and a Young diagram v E y, denote by
>
the restriction of the jet of P to the subset
We say that the jet is positive if (a) each of the following conditions: (1) C ai C 0, < 1; (2) ai > 0 for all i 1; or (3) pi > 0 for all i 1, implies P ( a ; P) > 0, and (b) if P & ( a ; p) 0 for all a /" v, but P[,l(a; /3) $ 0, then P[,l(a; P) > 0 for all ( a ; R) E Ak,l.
>
-
>
+
THEOREM 15 ([128], 58.1). A vector of real numbers fm(X), X E y m , is virtually positive if and only if the jet of the polynomial (6.3) is positive. 2.7. Branching of Macdonald polynomials. In Section 1.6 we regarded the Young graph as the branching graph of the Schur functions. By using other bases in the ring of symmetric polynomials R we can obtain new interesting branching graphs, including the - graphs which describe the theory of characters of the spin-symmetric group 6, (the group which linearizes the projective characters of the symmetric group 6,) and the branching of conjugacy classes of the symmetric groups 6,.
52. CHARACTERS OF 6,,
AND T H E YOUNG GRAPH
25
In the second chapter we will show that the multivariate Selberg's integrals can be computed as the Poisson integrals representing special harmonic functions on the branching graphs of Jack symmetric polynomials. An interesting class of symmetric polynomials was introduced recently by Macdonald [139, 1401 as a generalization of the more classical Hall-Littlewood polynomials. The Macdonald polynomials Px(x; q, t ) depend on two complex parameters q, t and are determined by the following properties: (a) the expansion of PAin the basis of monomial symmetric functions m x ( x ) ,
is determined by a matrix uxi, which is triangular with respect to the dominance ordering*) of Young diagrams A, p E y,; and (b) the polynomials PAare orthogonal, (7.2)
(PA,
P,),,t = 0
for
# p,
with respect to the scalar product that is defined on the basis consisting of Newton power sums by**)
form a linear For most values of the parameters q, t the polynomials {PA)XEy basis in the algebra R. The branching rule for these polynomials follows from Macdonald's results:
where the multiplicities z q t t ( X , A) are rational functions in the parameters,
Here the product is over all squares b of the diagram X that lie above the "new" square distinguishing the diagrams X and A in the same column. In our notation a = a(b) stands for the arm length of the square b, i.e., the number of squares in X that lie to the right of b in the same row. Similarly, 1 = l(b) is the leg length of b, i.e., the number of squares that lie below b in the same column. Figure 5 shows several first levels of the branching graph y ( q , t) of Macdonald polynomials; it differs from the ordinary Young graph only by the multiplicities of edges. The true Young graph y is obtained as the special case when q = t. If the parameters q , t are real and -1 < q, t < 1, then all multiplicities N ~ , ~ ( XA), , X 7A, are positive, so it makes sense to pose the problem of describing the boundary of the graph y ( q , t ) , i.e., of computing nonnegative harmonic functions on Y(q, t). *) See [49], 51, for the definition. **)We use the standard notation zx = 1'1 r ~2 T!2 r2!. . . , where r, = r,(X) is the number of rows of length i in a Young diagram X E y.
0.
INTRODUCTION
FIGURE 5. Branching of Macdonald polynomials. The extremality criterion from Section 1.6 reduces this problem to the following one.
The generalized problem of total positivity. Find all homomorphisms + R of the algebra of symmetric polynomials R that are nonnegative on Macdonald polynomials:
4: R
(7.6)
$ ( P A )2 0 for all
X E y.
If q = t , the polynomials PA reduce to the Schur functions, Px(x; q, t ) = sx(x), and the problem is completely solved by Thoma's theorem from Section 3.1. In Chapter 2 we will give arguments in favour of the following possible solution of the generalized problem of total positivity.
The formal generating series C O N J E C T U R[119]. E
52. CHARACTERS O F G,,
for homom~orphis~ns 4:R form
+
AND T H E YOUNG GRAPH
27
E% which are positizle in the sense of (7.6) are of the
where
This conjecture is completely proved in the following three cases: 1) for q = t (Schoenberg -Edrei theorem [84]); 2) for q = 0, t = 1 (Kingmall [129, 1301); 3) for q = 0, t = -1 (Nazarov [51]). The latter two examples are related to the classical Hall-Littlewood polynomials ([49],Chapter 111))which are the special cases of Macdonald polynomials when q = 0. In this case the multiplicity of an edge (A, A) is equal to
where r = r(A) is the number of rows of length j in the diagram A, arid j is the length of the row that conta.ins the square b = A \ A. 2.8. The characters and dimension group of a Kingman graph. When q = 0, t = 1 the Macdonald polynolnials reduce to the monomial sylnlnetric functions, (8.1)
Px(z;0 , l ) = nzx(x),
X
E
Y
The coefficients in Pieri's formula
for monomial symmetric functions are positive integers, equal to (8.3)
x(X,A) = rj (A):
where the multiplicities r, (A) are defined in the footnote to Section 2.7. Figure 6 shows several first levels of the corresponding branching graph. In this case the coherence condition for a family of probability distributions M, on the levels y, takes an especially natural form: if we begin with a random diagram A E y, distributed according to Mn(A), pick a uniformly distributed square of this diagram, and remove the rightmost square of the row containing this square, then the resulting diagram X E ynPlis distributed according to hl,-l. Kingman [129, 1301 found the boundary of this graph. His work was motivated by the analysis of the Ewens distribution on the symmetric group G,, which is very popular in populational genetics. We call K: = y ( 0 , 1) the Kingman graph. Handelman [log] arrived at the Kingman graph in a completely different way. He observed that if diagrams X and A have at most m rows, then the coefficient x(X,A) coincides with the multiplicity of the irreducible representation . i r ~of the group SU(7n) in the decomposition of the tensor product of .irx and the natural representation .ir(l) of this group.
0.
INTRODUCTION
FIGURE 6. The Kingman graph. From the viewpoint of the symmetric groups B,, the graph K describes the branching of induced representations of this group. The author's paper [34]contains a description of two equivalent graphs: one of them reflects the branching of conjugacy classes of symmetric groups, and the other describes the branching of partitions of a finite set; they are considered in more detail in Chapter 2, where we also give an approximative proof of Kingman's theorem on the boundary of the graph K ,and compute, for the first time, the semifinite traces and dimension group of this graph. Let us state our main results.
THEOREM 16 (on the Poisson integral). Denote by A the simplex of nonincreasing sequences
with y = 1 - C ai > 0. Define an embedding i: y + A by setting i(X) = ( X l / n , Xz/n,. . . ) for X E Y,. Let the Poisson kernel @(A;a ) be equal to the extended monomial symmetric function m x ( a ; 0; y ) given by
52. CHARACTERS O F G,,
AND T H E YOUNG GRAPH
29
T h e n the implications of the theorem of Section 1.3 hold; in particular, each harmonic function cp E Harm(K) has a unique integral representation of the form
where p is a probability measure o n the simplex A. THEOREM 17 (on semifinite characters). Extreme semifinite nonnegative harmonic functions o n the K i n g m a n graph are of the form
where H , is the differential operator, i n the ring R of symmetric functions, which is conjugate t o the multiplication by the complete homogeneous symmetric function h,, u i s a nonempty Young diagram, and the number of nonzero frequencies in the sequence Q = ( a l , C Y ~ ., . . ) E A i s finite. Let us define the jet of a symmetric polynomial P
E
R as the family of functions
a c A if the conditions for all ( a ; 0; y) E a and a /'
We say that the jet P is positive on a subset cpu(a;0; y) z 0
u
and imply cpu(a;0; y)
> 0 on
a.
$&(a;0; 7 )
#0
on
a
THEOREM18 (on the dimension group). Let P be a symmetric polynomial. T h e following conditions are equivalent: (a) There exists a number n such that all coeficients c~ in the expansion
are nonnegative. (b) The jet of the polynomial P i s positive o n the following subsets of the boundary A for all k = 1 , 2 , . . . :
2.9. Generalized Macdonald polynomials and orthogonal polynomials. A considerable part of Chapter 2 is devoted to studying the generalized Macdonald polynomials. The definition of these polynomials reproduces the definition of the ordinary Macdonald polynomials from Section 2.7 except for the following two points: (1) In formula (7.1) of Section 2.7 the matrix u x P is triangular not with respect to the dominance ordering, but with respect t o a certain total ordering of the set of Young diagrams, which is comparable with the dominance ordering. (2) (7.3) is replaced by a more general bilinear form
30
0.
INTRODUCTION
where w = (wl, w2,. . . ) is an arbitrary sequence of complex numbers, and . ... w,l, = wxl The generalized Macdonald polynomials will be denoted by PA(x;w). Our results may be divided into three parts. The theorems of the first group show that a number of properties of Macdonald polynomials, which Macdonald proved by special techniques, also hold, appropriately stated, for much more general polynomials PA(x; w). An example is provided by the following duality theorem.
THEOREM 19 ([119], Theorem 1). Denote by Qx the symmetric polynomials such that (Qx, P,), = SA,, and let P A be the generalized Macdonald polynomials associated with the conjugate ordering of Young diagrams,
Define a n automorphism w,
of the ring of symmetric functions R by setting
Then
where l / w = ( l / w 1 , 1 / ~ 2 ,... ) . The second group of results characterizes the original Macdonald polynomials in the class of the generalized ones. For example, Macdonald established the following property of his polynomials (i.e., corresponding to w, = (1 - q7') / (1 - t n ) ) : (SO) If X = ( X I , . . . ,An) is a Young diagram with n nonzero rows and :r = (21,. . . , x,, 0 , . . .), then (9.5)
PA(x; w) = 21 2 2 . . . 2, PA* (2;w),
where A, = (A1 - 1 , . . . , A n - 1). We show that this property does not hold for more general sequences w.
T H E O R E20 M ([119], Theorem 2). If property (SO) h,olds for n defining sequence is of the form
=
2, then the
where c is a constant, and the sequence {w,} is given by one of the following formulas:
(9.8) (9.9)
wn=(Y, wkm = a ,
TLXl,2, . . . , W, = 1 if n $ 0 mod rn.
Note that (9.8) and (9.9) are limit cases of (9.7). The results of the third kind establish a relation of Macdonald polynomials Px(x; q, t ) to the family of Rogers-Ramanujan orthogonal polynomials [154].The latter are obtained as follows. Substitute two variables x = ( x l , 2 2 ) into Px(x; q, t ) , then set 21x2 = 1, and consider PAas a polynomial in one variable y = xl r 2 .
+
53. T H E PLANCHEREL MEASURE O F
6,
31
This observation allows us to relate the characterization of the true Macdonald polynomials among the generalized ones to an old problem of Fejkr's [go]. This problem is stated as follows: among all polynomials of the form
where bo, bl, . . . is a scalar sequence, find orthogonal polynomials with respect to a certain measure. Fejkr's problem was essentially solved in [60] and [48]. We establish the equivalence of these two problems, and correct some inaccuracies contained in [60] and [48] (the case (9.9) in o m notation was overlooked there). The solution is based on the following unexpected fact.
THEOREM 2 1 [119]. The general solution of the infinite system of nonlinear equations
in variables f l , f 2 , f 3 , . . . depends only on three arbitrary parameters: if f 2 and f3 # f l , then fn = f l ( f 2 - f l ) ~ , , where
+
and Un(cos ip) = sin(n kind.
+ 1)ip / sin ip are the
# fl
Chebyshev polynomials of the second
In conclusion, we would like to draw the reader's attention to Theorem 6.3 from [119], which provides a new determinantal representation of certain Macdonald polynomials Px(x; q, t). These formulas were apparently not known even in the case of mono~nialsymmetric functions, i.e., q = 0 and t = 1. 53. The Plancherel measure of 6,
In Chapter 3 we will study the most interesting character of the group 6,, the character of its regular representation. This is simply the &function supported by the unity; however, its dual description in terms of the branching graph leads to the very important Plancherel growth process of Young diagrams. Our main result in the third chapter is a central limit theorem for the Plancherel measure. 3.1. Definition of the Plancherel measure. The Plancherel measure of a finite group G is*) a probability distribution on its dual object**) whose weights are proportional to the squared dimensions of irreducible representations. It is clear from Burnside's formula
*)See, e.g., [43], Section 12.4. **)I.e., on the finite set of equivalence classes of irreducible unitary representations.
32
0.
INTRODUCTION
that the weights of the Plancherel measure are equal to
Plancherel's theorem for the ordinary Fourier transform
claims that
(where tr(a) = C aii is the normalized trace of a matrix a E Mn@),and the Fourier inverse can be written in the form
A
If f , g are central functions, then f (n), ?(n) are scalar matrices, and it is convenient to regard them as scalar functions on G. Formulas (1.3), (1.4), (1.5) take the form A
A
where xT(w)/dimn is the normalized character of n E G. In particular, for w = e we obtain the identity
For an infinite discrete group G, there is also a standard definition of the However, Plancherel measure as a probability distribution on the dual space for wild groups similar to 6, this definition is not interesting. We suggest a new definition of the Plancherel measure for the class of locally finite groups. Fix an approximation G, = 1 4 G n of a locally finite group G, by finite subgroups G,, and let r be the corresponding branching graph.
e.
DEFINITION. The Plancherel measure of the group G, (with respect to the chosen approximation) is determined by the Markov chain on the branching graph r with transition probabilities JG,I p(A7A)= -. IGn+lI
x(A,A)dimn~ dim nx
'
53. T H E PLANCHEREL MEASURE OF 6,
where X E I', A E I',+l, and X / A. The initial state is the unique vertex 0 This Markov chain will be called the Plancherel growth process.
33
E
Fo.
This Markov chain is well-defined, as follows from counting the dimension of the induced representation
in two ways. We obtain
Gn+ll
lGnl
dim aA=
C
x(~ A), dim A,
A: A/A
whence CAp(X, A) = 1. Denote by Mn the distribution of the state X E r, of the Plancherel growth process after n steps. It is easy to check that Mn coincides with the Plancherel measure for G,,
which explains the choice of the term. 3.2. The typical shape of large random Young diagrams. The Planchere1 measure of the infinite symmetric group 6, is determined by the Markov chain on the Young graph with transition probabilities
This is an ergodic central measure on the space of infinite Young tableaux T; it corresponds to the zero frequencies a1 = a 2 = . . . = Dl = p2 = . . . = 0. Important results on the asymptotics of the shape of Young diagrams in the course of the Plancherel growth process were obtained by Vershik and the author [12], [18],and independently by Logan and Shepp [138]. Most problems considered in this book are motivated by these results. It will be more convenient to discuss the limit shape of Young diagrams if we slightly modify their original description. Following the combinatorial tradition, we have been believing up t o now that Young diagrams are merely graphic representations of partitions of positive integers*). Now we will change our viewpoint and regard Young diagrams as piecewise linear functions (Figure 7).
FIGURE7. Two ways t o draw a Young diagram *)Another generally accepted way of representing partitions, Ferrers diagrams (see [49], p. 17), is completely useless for our purposes.
34
INTRODUCTION
0.
DEFINITION.A Young diagram is any continuous piecewise linear function v = w(u) that has the following three properties: (1) wl(u) = f1; (2) w(u) = lul for sufficiently large lul; (3) the points of minima X I , .. . , xd and the points of maxima yl, . . . ,yd-1 of w are integers. For example, a one-square Young diagram w E yl corresponds to the function w(u) = max(lu1, 2 - l u l ) In general, the number of squares of a diagram w E yn can be written in the form
If we want to obtain a nontrivial limiting function for growing Young diagrams, we should uniformly rescale them t o keep the square of the subgraph (2.2) equal to one. Consider the mean value of rescaled Young diagrams with n squares with respect to the Plancherel measure Mn:
THEOREM 22. The uniform limit R(u) = lim wn(u) of the means (2.3) exists, n-03 and it coincides with the function
Since R1(u) = $ arcsin ,; we will call R(u) (as well as every statement which claims the uniform convergence to this function) the arcsine law. The asymptotic behaviour of the Plancherel means can be deduced from the following strong law of large numbers. THEOREM23 [12]. The uniform limit 1 lim --wn(u&)
n-w
+
= R(u),
where R is the arcsine law (2.4)) exists for almost all, with respect to the Plancherel measure, infinite Young tableaux t = (wl, wz, . . . , w,, . . . ) E T . Figure 8 shows the graph of the function R along with a random 100-square Young diagram obtained by computer simulation of the Plancherel measure. A remarkable consequence of the theorems on the limit shape of Young diagrams is the solution of Ulam's problem [I631 concerning the length Ln(x) of the longest increasing subsequence x = {xj),R=,. COROLLARY [12, 181. Let x be a sequence of 2.i.d. random variables with a common continuous distribution. Then, for every E > 0, (2.6)
{
lim Prob x:
n-00
IL$'
-21
-
<&}
=I.
53. T H E PLANCHEREL MEASURE OF GO,
35
FIGURE8. The arcsine law The proof of this statement uses an equally remarkable Robinson-ShenstedKnuth algorithm (see [152, 157, 1331, and also [45]), which can be regarded as a combinatorial version of the Fourier transform for the group G,. See [27] for further details on the applications of this algorithm to the theory of characters of the infinite symmetric group. The arcsine law also holds for the Young diagrams of maximum dimension. THEOREM24 [18].Assume that the maximum value of the function d ( w ) = dim^, on the set y, is attained at a Young diagram w,. Then
1 lim -w,
fi
( u h )= R(u).
3.3. The arcsine law for the distributions of tensors into symmetry types. The symmetric group 6, acts by permutations of factors in the space Cm of tensors of rank n over Cm. The unitary group U ( m ) is V,,, = also represented in the same space. The space V,,, breaks into the direct sum of subspaces V,,,(X) which are primary with respect to each of these actions and irreducible with respect to the joint action of the group 6, x U(m):
Here X ranges over the set y,,, of Young diagrams with n squares and at most m rows. The tensors from the subspace V,,,(X) are said to have symmetry type A. Denote by M'n,?n(A) =
dim Vn,m(X) dim V,,, , XEYn,,,
the relative dimension of the primary component V,,,(X). The weights h/ln,,(X) determine a probability measure on y,,,, the distribution of tensors into symmetry types. It turns out that this distribution obeys the arcsine law as n , m + oo.
0.
36
THEOREM25 [31].Given E
INTRODUCTION
> 0, consider the set of diagrams
that are, scaled by fi,uniformly close to (2.4). Assume that n, m lim $ = y > 0. Then
+
cc so that
In the case when n + cc and m is fixed, the asymptotics is completely different. Let us define the reduced row lengths of a Young diagram X E yn,, by
Then the vector x = ( x l , . . . , x,)
belongs to the cone
26 [31].Assume that m is fixed and n + oo. Then the joint distriTHEOREM bution of the reduced row lengths with respect to the measure Mn,, on y,,, weakly converges to an absolutely continuous measure Mm on the cone Cm with density
where
is a normalization constant. The standard argument which uses the electrostatic interpretation of the zeros of classical orthogonal polynomials*) yields the shape of the most probable diagram with m rows.
Let m be fixed and n COROLLARY.
+
cc. Denote by A(") the diagram from ( ~are ) the most probable. Then
y,,, such that the tensors of symmetry type x lim
n-cc
~j;")
-
n/m
fi
-
zk
-
Jm'
k=1,2
,... ,m,
where 2 1 , . . . , zm are the zeros of the Hermite polynomial
*)See [56],Section 6.7).
53. T H E PLANCHEREL MEASURE OF 6,
37
3.4. Gaussian limit for the Plancherel measure of the symmetric group. Denote by X X the character of the irreducible representation of the group 6, associated with a Young diagram X E Yn, and let X: be its value on the class of conjugate permutations with cycle structure p = (Ir1,2rZ,.. . ). For a fixed p E y,, the character is a random variable defined on the set Y, with the Plancherel measure M,. We are interested in the asymptotic behaviour of the distribution of this variable as n --, co. To obtain a nontrivial limiting distribution, we must normalize the values of characters in an appropriate way. Consider the random variables
corresponding to the values of characters on permutations with a single nontrivial cycle of length k = 2,3,. . . .
THEOREM 27 [121]. For any x2,x3, . . . , x,
E
R there exists the limit
lim Mn{X E Y,: cpk(X) < x k , 2 L k I m )
n-m
This result can be regarded as a central limit theorem for the distributions of characters of the group 6, with respect t o the Plancherel measure. It means that the random variables (4.1) are asymptotically independent and have Gaussian limiting distributions with zero mean and variance k. Formula (4.2) is a refinement of the theorem from Section 3.2 on the limit shape of the typical Young diagram with respect t o the Plancherel measure (which in turn may be compared with the law of large numbers). It describes the limiting Gaussian process for the deviations of a random diagram from its limit shape. A precise statement (equivalent to the previous theorem) is as follows.
THEOREM 28 [121]. Consider the random function
which describes the deviation of a Young diagram w E y, from the expected limit shape (2.4). A s n + a,the random process (4.3) converges i n probability to the Gaussian random process
where urn(2cos8)=
+
sin(m 1)8 sin 8
,em,.
are the Chebysheu polynomials of the second kind, and &,. . . . . are independent standard Gaussian variables. The covariance function of the process (4.4) is (4.6)
B (2 cos cp, 2 cos $)
=
1 log (sin lcp + $112) 2sincpsinG sinlcp-$112
-
38
0. INTRODUCTION
New interesting restatements of the central limit theorem can be obtained by considering other functionals on Young dagrams. Denote by t,(x) the orthogonal Chebyshev polynomials of the first kind, (4.7)
t , (2 cos 19) = 2 cos mI9,
and let
where ( f l , . . . , f d ; 91,. . . , g d ) are the Frobenius parameters of a diagram A, see Section 2.4.
THEOREM 29 [121]. The Plancherel mean value of the random variable t , equals
for odd indices, and vanishes for even indices. A s n + cm, the centralized random = t,(X) - (t,)%, X E yn, are asymptotically independent and variables :,(A) normal. The limiting variance o f t , is equal to m .
Recall that the content of the square-b = ( i ,j) of a Young diagram X is the number c(b) = j - i . The collection of contents of all squares of a diagram uniquely determines this diagram; thus it can be considered as another system of parameters. Let
where urn are the orthogonal Chebyshev polynomials of the second kind (4.5).
THEOREM 30 [121]. The Plancherel mean value of the random variable urn equals
for even indices, and vanishes for odd indices. A s n -, cm, the centralized random variables ;,(A) = u,(X) - (u,),, X E yn, are asymptotically independent and normal. The limiting variance of urn is equal to l / ( m I).
+
The proofs of the above theorems are based on studying the grading, on the algebra of conjugacy classes of finite symmetric groups, introduced in [121]. A complete description of the convolution of conjugacy classes of symmetric groups is a difficult combinatorial problem, see [89], [ l o l l . 3.5. The h o o k w a l k a n d its q-analogues. Greene, Nijenhuis, and Wilf [102], [I031 found a remarkable combinatorial algorithm which they called the hook walk. The algorithm was intended for two purposes: for.proving the hooklength formula (see Section 3.6, formula (6.12)), and for generating random Young
53. T H E P L A N C H E R E L M E A S U R E OF 6-
39
tableaux distributed according t o the Plancherel measure. More precisely, the algorithm allows one to simulate the transition probabilities of the Plancherel growth process. The original algorithm is as follows. Put a Young diagram A into a sufficiently large rectangle II (Figure 9). Start with the rightmost lowest square of the rectangle and choose (uniformly) an arbitrary square of the complement II \ X that lies either above the initial square in the same column, or to the right in the same row. After several iterations of this procedure the process will stop a t one of the external corner squares b of the diagram A. Denote by A the Young diagram obtained by attaching the square b to A. Then the probability of the transition X /" A coincides with the transition probability of the Plancherel growth process: Prob {A /" A) =
(n
+d(A) 1) d(X) '
A
E
Y,.
FIGURE 9. The hook walk. It is shown in [36] that the hook walk algorithm can be adapted to generate more general central measures. It still seems somewhat mysterious that exactly the same class of central measures can be used for constructing topological invariants of knots in the Jones-Okneanu scheme, see [114], [93]. The link is provided by the characters of the infinite-dimensional Hecke algebra which satisfy the A. A. Markov (Jr.) property. The description of all such characters in terms of frequencies was obtained by Vershik and the author [19]. A central ergodic measure ~ " 1 0on the Young graph is called [36] a knot measure if its frequencies (Thoma parameters) a, P satisfy one of the following conditions: (a) the frequencies al, a 2 , . . . form a finite or infinite geometric progression with unit sum, p 0; (b) the same with a and p interchanged; (c) the frequencies a l laz, . . . and Dl, p 2 , . . . form a pair of infinite geometric progressions with a common exponent 0 < q < 1 and unit sum C ( a i +Pi) = 1; (d) a = 0, p 0 (the Plancherel measure). In Chapter 3 we describe a generalized hook walk algorithm which generates all knot central measures. At t,he same time we obtain the following q-analogue of the hook-length formula from [92].
-
--
40
0.
INTRODUCTION
THEOREM 31 1361. Attach to the edges of the Young graph y the "multiplicities"
and recursively define the "dimensions" by setting d,(@)= 1 and (5.3) Then
+ + +
where [k], = 1 q . . . q"', [k],! = [I], [2],, . . . , [k],, h(b) is the hook length *) of a square b in a diagram A, and n(A) = (k - 1)Ale.
xk21
For q = 1 the multiplicities are equal t o one, and (5.4) reduces to the ordinary hook-length formula. 3.6. Random Young tableaux and Selberg's integrals. Selberg ([156], see also [69]) computed certain multivariate integrals, including
This integral and some similar ones**) are discussed in [151]and [66]. We suggest a new approach to computing Selberg's integrals. Its key point is the interpretation of these integrals as the Poisson integrals (3.5) from $1 which represent special harmonic functions on the Young graph and its deformations. Let us illustrate this idea on a simple model example with the Euler beta integral
It is easy t o check that the function
+
+
where (c), = c(c 1). . . (c n - 1) is the Pochhammer symbol, is a harmonic function on the Pascal triangle (see Section 1.4). Like all harmonic functions, it *)The hook length is the number of squares to the right of b in the same row and below b in the same column, counting b itself once. **)We call them Selberg's integrals.
§3. T H E PLANCHEREL MEASURE OF G,
41
can be represented by the Poisson integral (3.5), $1. The density of the representing measure can be obtained as the "radial limit" lim n Mn ( [np], n) =
n-00
r ( A + B) J3A) r'(B)
(1
of the probability distributions
and (6.2) follows immediately. In order to obtain Selberg's integrals, replace the Pascal triangle by the truncated Young graph Y(k) and its deformations. By definition, the vertices of Y(k) are Young diagrams with a t most k rows, the edges being the same as in the complete Young graph Y. Instead of the coherent system of distributions (6.5), consider the family*)
depending on a parameter A > 0. We will verify that these distributions correspond to the transition probabilities
so that (6.6) defines a coherent family of probability distributions on y ( k ) . Letting n + oo, with the assumption that
we find the density of the representing measure
where Ck is a normalization constant,
The general theorem on the Poisson integral (Section 1.3) allows us to conclude that
J.1
sx(al,
..,
1ji<j
-1, . - ,ek?O e l + . .+uk=l
(6.11)
*)We denote by c(b) = j
n
k
lai
-aj12na;.-1dal j=1
k unities
- i the
contents of a square b = (i,j)E A.
...dak-l
0.
42
INTRODUCTION
where s x ( a l , . . . , cuk) is the Schur f ~ ~ n c t i owhich n determines the Poisson kernel in the case under consideration. The most nontrivial part of the above argument, the choice of the transition probabilities (6.7), is a simplified version of the central topic of the paper [125], which deals with infinite-dimensional analogues of (6.11). A typical property of the distributions (6.6) is their multiplicativity: the weight of a Young diagram X E y n ( k ) is the product of simple factors associated with separate squares b E A. The Plancherel measure also enjoys this property. Indeed, using the famous hook-length formula [92]
for the dimensions of irreducible representations of the group plicative form for the weights of the Plancherel measure:
en,we find a multi-
Here h(b) is the hook length of a square b E A, as defined in the footnote to Section 3.5. The most general form of Selberg's integrals obtained in this book is
J . . J.
P A ( a l ,.
.,ak;~)
l
el,.. , a k > O el+ ak=l
(6.14)
n
...+
k
lai - a j ( 2 0 , a f - 1 d ~ i . . . d a k - l
j=l
k unities
Here Px(x; 0) are the so-called zonal symmetric polynomials*), which arise as limiting cases of Macdonald polynomials from Section 2.7. The proof uses the truncated branching graph of zonal symmetric polynomials in finitely many variables X I , . . . , xk (Figure 10). The zonal symmetric polynomials were studied in [ I l l , 112, 159, 1061. The integral (6.13) itself is not new (see [151],though it contains no proof). The limiting case of (6.14) when 0 = 0 and k -+ co,A -+ 0, so that kA -+ t > 0, is worth special mention. In this case (6.14) takes the form**)
where pt is the well-studied Poisson-Dirichlet distribution on the infinite-dimensional simplex A which is the boundary of the Kingman graph (8.4) from Section 2.8. *)Also known as Jack polynomials. **)As above, r3 is the number of rows of length j in A, and 1 ( A ) = of rows of A.
rj
is the total number
$4. CONTINUOUS Y O U N G DIAGRAMS IN PROBLEMS O F ANALYSIS
43
FIGURE 10. Branching of zonal symmetric polynomials. $4. Continuous Young diagrams in problems of analysis
The fourth and final chapter of the book deals with applications of Young diagrams and their continuous analogues in various problems of analysis and probability theory. The key idea of these applications consists in systematically studying, along with ordinary Young diagrams (regarded as piecewise linear functions), their uniform limits, called continuous diagrams. We will show that many constructions and algorithms, typical for the combinatorics of Young diagrams, can be extended by continuity and applied to arbitrary continuous diagrams. In this book we consider in detail the transform that associates with a Young diagram its transition probability in the Markov process of Plancherel growth. We obtain a limiting analogue of this transform which establishes a nontrivial bijective correspondence between continuous diagrams and arbitrary probability distributions. This correspondence, which seems rather strange at first sight, in fact arises in a number of problems, such as the continuous version of the partial fraction expansion, computation of integrals with respect to random Dirichlet measures, and the Markov moment problem. In connection with moment problems, this bijection
44
0.
INTRODUCTION
(in different terms) was used by M. G. Krein (see [47]), so we call it the Krein correspondence. In Section 3.5 above we mentioned a beautiful combinatorial algorithm, the hook walk, intended for the stochastic simulation of the transition probabilities of the Plancherel measure of the infinite symmetric group 6,. Here we will find its limiting version, the interval shrinkage algorithm, which can be applied to arbitrary continuous diagrams. This algorithm provides a new description of the Krein algorithm in probabilistic terms. The second important topic of Chapter 4 is the asymptotic behaviour of interlacing sequences. It turns out that in many situations (separation of roots of orthogonal polynomials, spectra of large random matrices, etc.) the diagram describing a pair of interlacing sequences obeys the same "arcsine law" as large random Young diagrams (see Section 3.2). Some light on the reasons for the universality of the arcsine law is shed by free probability theory [167], where this law (more precisely, the closely related semicircle law) plays the same role as the normal distribution does in traditional probability theory. We show that the continuous limit of the Plancherel growth process is the simplest (semicircle) diffusion, in free probability theory, governed by the Burgers equation Rt R R, = 0.
+
4.1. The Plancherel measure and the partial fraction expansion. Let us use the definition of Young diagrams as piecewise linear functions, see Section 3.2. Recall that we denote by X I , x2,. . . , xd the points of minima, and by yl, . . . , yd-1, the points of maxima of a diagram v = w(u). Condition (3) is immaterial for what follows, so we drop it. A function v = w(u) that satisfies only the two conditions (1) wl(u) = &I, and (2) there exists c E R such that w(u) = lu - cl for sufficiently large / u ( , will be called a rectangular diagram. The point c = C x k - C yk is called the centre of the diagram, and the number
the area of the diagram. As a Young diagram X grows, the abscissa of the new square must coincide with one of the minima xk, k = 1,.. . , d. The corresponding Plancherel transition probability p(X, A) from (2.1) in $3 will be denoted by pk,
In [I171 and [36],the following simple formula*) for this probability was found:
The next theorem is an immediate, but important, consequence of this formula. *) Some versions of (1.1),less convenient for our purposes, were published earlier in [ll]and [441.
$4. CONTINUOUS YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
45
THEOREM 32 [118, 371. Let u = X(u) be the Young diagram determined by a pair of sequences X I , 22,. . . , xd and yl, . . . , yd-1. T h e n its Plancherel transition probabilities (1.1) are precisely the coeficients of the partial fraction expansion
Note that formulas (1.2), (1.3) make sense for an arbitrary rectangular diagram w and determine a discrete probability distribution with atoms of weights pk at the points xk, k = 1,.. . , d. The coefficients in (1.3) will be positive if and only if the zeros and poles of this fraction interlace:
Formula (1.3) and the definition of the extended complete homogeneous symmetric functions h, from Section 3.4 imply a simple formula for the moments of the measure (1.1):
The first three moments have especially simple forms:
For true Young diagrams, h l = 0 and the variance h 2 = n depends only on the number of squares in the diagram (but not on its shape).
COROLLARY. Given A > 0 and k = 1 , 2 , .. . , set
T h e n CApY)(X,A) = 1 for all X E y , and the probabilities (1.7) determine a central Marlcov chain o n the k-row part Y(k) of the Young graph. Thus we have justified the derivation of Selberg's integral (6.10) in Section 3.6. The more general integrals (6.14) in $3 can be obtained in a similar way; in this case the transition probabilities are also given by (1.7)) but instead of the parameters xk, y k of the original Young diagram one should use the parameters xk(0),yk(0) of the rectangle diagram X(0) obtained from X by rescaling by 0 along one of the axes. This is one more argument in favour of taking a serious attitude toward nonintegral rectangular diagrams.
4.2. Asymptotic behaviour of interlacing sequences. How can a pair of interlacing sequences
behave as n grows? In order to make the question more precise, consider the rectangular diagram w, with minima points {xk) and maxima points {yk); it is easy to see that this diagram is uniquely determined. We can now restate the
46
0.
INTRODUCTION
problem as follows: what is the lirnit shape of the diagrarn w,? We saw in $3 that the lirnit shape of typical Young diagrams with respect to the Plancherel measure is described by the function
+d
arcsin
m )
if
lu1 5 2,
if
lu1
> 2.
Here we will show that the sarne "arcsine law" holds in purely analytical situations which seem cornpletely unrelated to the symmetric group. In our first example we consider a family of orthogonal polynomials {Pn(u)}r=o defined by a linear recurrence relation
and initial conditions Po(u) = 1, PI(u) = u the adjacent polynornials
-
bl. It is well known that the roots of
interlace. To describe the character of mtittial separation of the sequences {xk), {yk), we use the rectangular diagram w, defined by the following conditions: (2.5)
P n 1(u) wk ( u ) = sign , wn(u) = Iu P n (u)
-
Cn 1
for sufficiently large lul,
where Cn = C - C y,. It turns out that for a wide class of orthogonal polynomials (including the classical polynomials of Jacobi, Laguerre, arid Herrnite) the asymptotic behaviour of interlacing roots obeys the arcsine law. THEOREM 33 [38]. Assum,e th,at th,e coeficients of th,e recurrence relation (2.3) satisfy th,e conditions
Then th,e linrit 1 lirn -w,(~c,-~ cn
n-oo
+ b,)
=f~(u)
exists uniform,ly in u E EX, th,e function fZ being given by (2.2) Figure 11 shows the diagrams of root separation for the Chebyshev, Hermite, and Laguerre polyriornials of degrees 15 and 16. The graph of the function (2.2) (appropriately rescaled) is laid upon the diagrams. The asymptotics of the separation of roots should be distinguished from the asyrnptotics of their distribution. For example, the lirnitirig density of roots of the Jacobi polynomials is described by the inverse semicircle law, 1 lirn -#{i: xi
n-oo
n
< u)
=
dn: -
IuI
I 1;
54. CONTINUOUS YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
-1.0
-0.5
0.0
0.5
47
1.0
Chebyshev polynomials
Hermite polynomials
Laguerre polynomials FIGURE11. Separation of roots of orthogonal poplynomials.
and the same density for the Hermite polynomials is described by the semicircle law,
Thus the limiting densities of roots are different, but the arcsine law holds in both cases. It is not difficult t o give sufficient conditions for the arcsine law in terms of the measure ,u with respect to which the polynomials P, are orthogonal. For example, the law holds if the measure is absolutely continuous on [-I, 11 and the function log,ul(cos0) is summable on [0, T I . The second example of an appearance of the arcsine law in an analytical problem is related to Wigner's classical semicircle law (see [142], [55]), which claims that in the typical spectra of large random matrices the eigenvalues are distributed with the semicircle density 2d-l~. Consider a stable mechanical system with n degrees of freedom which oscillates near the equilibrium. Its potential energy is quadratic and determined by a symmetric positive definite matrix A. The squares of the fundamental frequencies of the system coincide with the eigenvalues X I ,. . . , x, of this matrix.
0.
48
INTRODUCTION
If a linear coristrairlt is imposed on the systern, the11 the new fundamental frequencies interlace with the initial ones, so that the separation conditiorl (2.1) holds. This result is know11 as Rayleigh7s theorern (see [I],$24). The rectangular diagram w, that describes the interlacing spectra of an Hermitian matrix A and its restriction to a certain hyperplane h will be called tlle rigidity diagram. THEOREM34 [38]. Let w, be the rigidity diagram of a random symmetric matrix A(,) of order n, with respect to a random hyperplane 11. c I%,. Assume that the normal to h is uniformly distributed on the unit sphere ofRn, and the matrix ( entries ai:), i j , are independen,t of 11. an,d among themselves, and identilnlly
<
distributed with mean value ~ ( a j ; ) )= 0 and variance ~ ( a j ; ) )= 1. Then lim n+,
1 --EW,
J5;
( u 6 )= f l ( 7 ~ )
uniformly in u E R. Recently Voiculescu (see [167]) obtained a conceptual proof of Wigner's theorem. He and his pupils developed the so-called "free probability theory", where Wigner's theorem arises as a consequence of the "central limit theorem" in which the role of tlle Gaussian distribution is played by Wigner's serriicircle law. 4.3. Diagrams, Rayleigh measures, and their moments. DEFINITION. A contin,uous diagram (or sirriply diagram) is any function ZI = w(u) with the following two properties: (1) Iw(u1) w(u2)I Iu1 - u2I; (2) w(u) = lu - cl for some c E R and suificiently large lul. -
<
A typical example of a corltinuous diagrarrl is the f~inction(2.2). Denote by D = 2) [a,b] tlle space of diagrams supported by the interval [a,b] , i.e., the space of diagrarns with w(u) = lu cl for u $ [a,b]. We endow 2) with the uniform convergerlce topology; clearly, the subset of rectangular diagrarns VcDo is dense in 2). The point c is called the centre of the diagram w, and the number -
is called its area*). Our choice of diagrams for representing interlacing sequences is due to purely historical reasons (namely, it is motivated by the theorern on the lirnit shape of large Young diagrams); it is by no means the only possible way. Let us present another approach, which is closer to [47]. It is based on the notion of interlacing measures. DEFINITION.A finite u-additive function called a Rayleigh measure**) if (3.2)
r(-m,x)
>0
and
r(x,+m)
T
defined on Bore1 subsets of
>0
EX is
for all x E R.
*)Integrals without explicitly stated limits are assumed t o he taken over the whole real axis. **)The choice of the term is motivated by the above Rayleigh's theorem on the separation of spectra of a n Hermitian matrix and its restriction t o a hyperplane (see Section 4.2). Rayleigh's theorem for operators is closely related t o the notion of the spectral shaft function, see [6].
54. CONTINUOUS YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
We say that nonnegative Bore1 measures p, v interlace if the difference is a Rayleigh measure.
49 T
=p -v
Let us give two examples. ( 1 ) Each nonnegative measure T is a Rayleigh measure. ( 2 ) Let pn be the sum of &measures supported by points X I ,xz, . . . , x,, and let vn be the analogous sum for points yl, . . . , yn-1. The measures pn and vn interlace if and only if the separation condition (2.1) holds. It is easy to see that Rayleigh measures rn = pn - vn weakly converge to a limit T if and only if the limit w of the corresponding rectangular diagrams wn exists. Both limits are related by simple formulas:
(3.3)
def 1 r ( u ) = T { ( - m , '11)) = -2( 1
+wf(u)),
Figure 12 illustrates the geometric meaning of (3.4). Note that the function f used by Krein and Nudelman in [47] (Chapter VII, $2.1) is complementary to our Rayleigh measure:
f ( u )= 1 - T(u).
(3.5)
FIGURE
12. A diagram and its Rayleigh measure.
DEFINITION. The moments of a diagram w E its Rayleigh measure:
V are the ordinary moments of
In terms of the diagram itself, the moments are given by
We will need the moment generating function
$4. CONTINUOUS Y O U N G DIAGRAMS IN P R O B L E M S O F ANALYSIS
51
The correspondence between Rayleigh measures and probability distributions was discovered*) in [47] from quite different considerations. It was used t o establish a relation between two moment problems: the Hausdorff moment problem,
and the (0, 1)-Markov moment problem,
where T is a Rayleigh measure**). A result of Krein and Nudelman ([47], Chapter VII, 52, Theorem 2.1) claims that a sequence {p,} is a solution of the Markov moment problem (4.4) if and only if there exists a solution of the Hausdorff problem (4.3) for the sequence {h,) related to the original sequence p, by the identity
This formula coincides, up to notation, with (4.2). For example, if we take the probability measure with density (3.12) as a Rayleigh measure T, then p can be shown ([37],2.3, Example 1)to be the semicircle law
the moments (4.3) of this measure are the Catalan numbers***), popular in combinatorics:
(odd moments vanish), and both sides of (4.5) coincide with the function
This computation elucidates the appearance of the,"arcsine law" (2.8) near Wigner's semicircle law. The third approach to the Krein correspondence is related t o the random Dirichlet measures arising in nonparametric statistics (see [91]).As a simple example, consider a discrete probability distribution T with atoms of weights TI,. . . , T, at real points X I , .. . , x,. It determines the probability measure on the (n - 1)dimensional simplex
*)The author does not know any earlier reference. **)We modify the statement of the Markov moment problem and its solution from [47] in order t o make the terminology closer to ours. ***)See [164] for further details on the combinatorial theory of orthogonal polynomials.
52
0.
INTRODUCTION
with the Dirichlet density
Denote by p the distribution of a linear functional
the simplex A,. It is known (see [82]) that the original distribution T is related p precisely by the Krein formula (4.2). Analogously, we can associate (see [131],Chapter 9) with each probability measure T on a real interval a random Dirichlet process whose realizations are random probability measures. The expected value
X=
(4.11)
S
xda(x)
with respect to the random measure a is a random variable; denote its distribution by p. Then T and p are related by (4.2). An explicit formula for the density f of the random variable (4.10) in the case of a discrete measure T was found in 1821: f(x)
=
I sin
(. ) fi 1% Xk<X
-
xkTk
k=l
A more general formula (4.13)
y7(")
1 7rwf(x) 1 (z) = - cos 7r 2 J(b-x)(x-a)
exp
lb
1 dwo 2 u-x
for the density p(W)(x)= dp / dx in the case of an arbitrary probability measure T was conjectured in [37], (2.8.2). As it turned out later, it can easily be derived from the results of [80]. The paper 1371 also gives an expression for the purely discrete part of p in terms of the Rayleigh measure T ([37], (2.8.1)). The author does not know any explicit general formula for the distribution p in terms of T . 4.5. A differential model of growth of Young diagrams. The Plancherel growth of Young diagrams, which was considered in detail in $3, is a random process with discrete time and space. Using the Krein correspondence, we will construct a differential model of Plancherel growth, a nonrandom process with continuous time and space which however displays the same asymptotic properties as the Plancherel measure of the infinite symmetric group 6,. Our differential model is adequately described by the Burgers equation
The analogue of the arcsine law describing the shape of large random Young diagrams is the automodel solution
54. CONTINUOUS YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
53
of (5.1). We will show that in a certain sense all solutions are asymptotically close t o (5.2). Though the Burgers equation is well studied*), we did not manage t o find this fact in the literature. The idea leading to the differential model is very simple: instead of attaching t o a Young diagram X one random square with Plancherel transition probabilities (1.2), we wish t o attach the parts of this square proportional t o the Plancherel probabilities to all possible positions simultaneously. In order to implement this idea, replace the set of Young diagrams y with the space of all continuous diagrams 2). The analogues of Young tableaux are curves R+ 3 t H wt E D in the space of diagrams such that the functions wt(x) = ~ ( xt ), are nondecreasing in t. Without loss of generality, we may take the area (3.1) of the diagram as the parameter t on the curve**) wt. Then the partial derivative &w(x, t ) is the density of a probability distribution in z, which is conveniently regarded as a tangent vector t o the curve. On the other hand, the Krein correspondence associates with each diagram w E 2) its "transition measure" p ; thus we obtain a vector field on D. Equating the tangent distribution at a general point of the curve wt and the transition distribution of the corresponding diagram, we arrive at the basic dynamic equation. Unfortunately, since there is no explicit formula for the transition measure (Krein correspondence), we have t o write our equation using the characteristic identity (4.2): exp
(- 1
d -u(u, du
t)
-)
z du 1 - uz
=
1
d du ~ ( ut ), dt 1 - uz
-
(for sufficiently small 2 ) . Here we have used the auxiliary notation ~ ( u , t )= 1 z(w(",t) - lull. Equation (5.3) can be essentially simplified if we rewrite it in other variables. For example, for the coefficients h,(t) of the generating function
we obtain a chain of ordinary differential equations
It is even more interesting to rewrite (5.3) in terms of the function (5.4) itself. THEOREM36 [40]. The basic dynamic equation (5.3) for an indeterminate function ~ ( xt), is equivalent to the Burgers equation
for the function (5.4). Analyzing equations (5.5), we arrive a t the arcsine law as the common asymptotics of all solutions of (5.3) as t 4 oo. *)See, for example, E. Hopf's paper [110], which is entirely devoted t o this equation **)In this section we consider only diagrams centered at the origin, c = 0.
54
0.
INTRODUCTION
THEOREM 37 [40]. Assume that a function a ( x , t) = (w(x,t) - 1x1) / 2 satisfies the basic dynamic equation (5.3)'. Then
uniformly in x E R. The following result allows us to relate the growth of Young diagrams to the Burgers equation more directly.
FIGURE 13. Differential growth of a Young diagram.
{yk}i=i
Fix a Young diagram w E Y , and let {xk)$=,, be the points of its minima and maxima. Consider a one-parameter deformation wt of w by attaching a tiny square*) of area p k t , where pk is the Plancherel probability (1.2), above each minimum xk. Besides the former maxima yk, the resulting rectangular diagram wt (Figure 13) has new maxima at the points yk(t) = xk; and each minimum xk is Consider the rational fraction split into the pair x t ( t ) = xk f 6.
(coinciding with the left-hand side of (1.3) for w t ) . Then
i.e., R satisfies Burgers' equation with initial condition R(x, 0).
4.6. The Plancherel growth and the Voiculescu diffusion. The book [167] develops a nonstandard version of probability theory based on replacing the ordinary convolution with the free convolution p EE v of probability distributions p , V. Without going into details, we only mention that the distribution p BI v arises as the spectral measure of the sum of two self-conjugate operators, provided that *)Only small values o f t are of interest
54. CONTINUOUS YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
55
they generate a free algebra and have spectral measures p and u, respectively. For example, the semicircle distributions
form a semigroup with respect t o the free convolution:
Voiculescu's theory contains natural analogues of many notions and theorems of traditional probability theory. In particular, the role of the Gaussian distribution is played by the semicircle law, and the analogue (6.1) of the Gaussian diffusion semigroup is implicitly described by the Burgers equation (5.6) for the CauchyStieltjes transform
of the diffusing measures pt = p W ut, t > 0. Relating the Plancherel growth of Young diagrams and the Voiculescu diffusion by means of the Burgers equation allows us to obtain the diffusion equation directly in terms of pt:
where f (x) is a compactly supported test function. We emphasize that the original papers [166, 71, 1671 do not contain equation (6.4); their authors use only implicit descriptions of the semicircle diffusion. We also obtain direct information on the infinitesimal mass transfer in the Voiculescu diffusion.
T H E O R E38. M Let p be a discrete probability distribution with atoms of weights pk at points xk, k = 1 , 2 , . . . , n, and let pt = p W ut, where ut is the semigroup (6.1). Consider the family of discrete probability measures fit assigning the weights
to the points
where
Then the measures p t ,
for every polynomial f
fit coincide up to infinitesimals of order greater than t :
0.
56
INTRODUCTION
4.7. The interval shrinkage process. In Section 4.4 above we described three different approaches t o the Krein correspondence between continuous diagrams and probability distributions on the line. According to one of them, the most important for our work, the distribution ,u E M corresponding t o a diagram w E D should be understood as a generalization of the transition probability of a Young diagram in the Markov process of Plancherel growth. It is natural to expect that each description of the Plancherel measure of the infinite symmetric group 6, has an appropriate limiting analogue which determines the Krein correspondence. Here we will justify these expectations by constructing a limiting version of the hook walk algorithm from Section 3.5, the interval shrinkage process. The interval shrinkage process is governed by a given diagram w E D ,or, equivalently, by its Rayleigh measure r (see Section 4.3). In the course of this process an infinite chain of random nested intervals [ao, PO]3
(7.1)
[&I,
PI] 3 . . . 3 [ a n , Pn] 3
is constructed. With probability one the intersection of these intervals reduces to [a,, P,]. The main statement claims that the distribution a single point X = of the random variablg X constructed from the diagram w is exactly the Krein transform ,u of this diagram. A precise description of the interval shrinkage process is as follows. We begin Po] containing the support of the Rayleigh measure with an arbitrary interval [ao, 7. Each step of the process consists of two operations: (1) divide the current interval [an-1,,f3n-l] into two parts by a random point y uniformly (i.e., according to the Lebesgue measure) distributed on this interval, and (2) let the next interval [a,, p,] be one of these parts chosen a t random with probabilities depending on the measure r :
n,,,
(7.2)
[..,A1
=
{
[a,-1,
[y,
y] with probability r(-co, y], with probability r ( y , +co).
The process is well-defined, since, by the definition of Rayleigh measures*), the numbers r(-co, y], r ( y , +m) are nonnegative and sum t o unity. All choices are assumed t o be independent.
THEOREM 39 [37]. Assume that a chain of random nested intervals (7.1) is constructed from a Rayleigh measure r by the interval shrinkage process. Then the limits (7.3)
x = lim a,
= lim
P,
coincide with probability one, and the distribution ,u of the random variable X satisfies the Krein identity (4.2). As an illustration, consider the "triangular" Young diagram
The Rayleigh measure of this diagram is the probability distribution with atoms of equal weights r{+l) = 112 at the points u = +1. We divide the interval *)We consider only measures of unit total charge T(-co, cm)= 1.
$4. CONTINUOUS YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
57
[a,-l, Pn-1] into two parts [anpl, y], [y,Pnpl] by a random point y , and then let [a,, p,] be one of these two parts chosen with equal probabilities. It is easy to check that in this example the distributioi~of the random point (7.3) has the density
in agreement with (4.12). A link between the original hook walk algorithm and the interval shrinkage process is provided by the following procedure for simulating the coefficients of partial fraction expansions. Consider a rational fraction
with interlacing zeros and poles,
and positive coefficients pk (these two conditions are equivalent). We will describe a Markov chain that reaches one of the poles X I , . . . , x, in a finite number of steps, the probability of reaching xk being equal to the corresponding coefficient pk. The states of this chain are intervals of the form [xi,xj]. The initial state is the largest interval [xl,x,], and the transition to a new state is performed as follows. Pick a uniformly distributed point y on the current interval [xi,xj]. It falls into one of the subintervals formed by the numbers (7.7). There are two cases:
for some k. By definition, the new state is [xi,xk] in the first case, and [xk,x3] in the second case. At each step of the process the ends of the current interval approach each other. After at most n steps the interval degenerates into a point, and the process stops at one of the poles of the original fraction.
THEOREM 40 [37]. T h e probability that the above Markov chain stops at the point xk is equal t o pk.
CHAPTER 1
Boundaries and Dimension Groups of Certain Graphs In $1 of the Introduction we have briefly presented the theory of locally semisimple algebras. This theory consists of three principal components. Combinatorial methods for studying an LS-algebra 8, are based on choosing an approximating family {8,)~=, of finite-dimensional semisimple subalgebras and considering the branching graph associated with this family. These methods are especially good when there exists a naturally distinguished approximating family of finite-dimensional subalgebras. This is the case for the group algebra of the infinite symmetric group @[6,]. Note that the theory of LS-algebras provides a motivation for posing new problems in traditional fields of combinatorics. An example is formula (4.5) from $2 of the Introduction, for the number of skew Young tableaux of a given shape. Another class of methods is based on studying the Grothendieck group KO(%,) regarded as an ordered abelian group with a distinguished order unity. The advantage of this method is that it involves only the limiting algebra 8, and does not depend on the choice of a particular approximation. The third circle of ideas is based on applying the methods of the metric theory of dynamical systems. One can define a measurable transformation S (called the adic shift) in the path space of the branching graph so that the central measures of this graph are identified with S-invariant measures. This observation leads to the ergodic method, the most powerful and universal method for computing characters of LS-algebras. In this chapter we supplement the survey of the theory of LS-algebras given in $1 of the Introduction as follows. In $1we prove the ergodic method which we use for the computation of central measures for various branching graphs. $2 contains some examples of branching graphs of combinatorial origin related to the Young graph. They illustrate the notion of similarity of branching graphs suggested by the author in [34]. In $$3 and 4 we compute central measures and dimension groups for the branching graph of induced representations of the symmetric groups 6, and for the generalized Stirling triangles. $1. Ergodic method
In this section we prove the ergodic method for computing characters of LSalgebras which was stated in Section 1.5 of the Introduction. Consider a branching graph I?, that is, a directed graded graph satisfying conditions (a)-(d) from Section 1.7 of the Introduction. Let the edges (A, A) of this graph be equipped with positive (but not necessarily integral) multiplicities x ( A , A).
60
1.
BOUNDARIES A N D DIMENSION GROUPS OF CERTAIN G R A P H S
Denote by T the space of paths (1.1.1)
t
=
(Ao,XI,. . . , A,,
. ..)
>
in the graph starting at Xo = 0 E r o . By definition, /" A, for all n 1. We endow T with the componentwise convergence topology, in which it is compact. . . , A,) the product Associate with a finite path u = (A,,
of the edge multiplicities along this path, and let
where the sum is over all paths u from Am = X t o A, = A. In particular, if X = 0 we write simply d(A) = d(0, A) and call d(A) the dimension of the vertex A E I?. Denote by Cu C T the cylinder set consisting of all paths with a given initial interval u = (AO, XI,. . . ,An). A Bore1 probability measure M on the path space T is called central (with respect to the multiplicity function x) if the quotient
depends only on the endpoint A, = X of u. The function p: I' + R+ defined by (1.1.4) uniquely determines the central measure M and is characterized by condition (2.2) from Section 1.2 of the Introduction. The description of all central measures for a given pair (I',x) is one of the main problems in the theory of LS-algebras. For example, the description of characters of locally finite groups can be reduced to this problem. The most important are ergodic central measures, defined in Section 1.5 of the Introduction. It is easy to see that if the limits p(X) = lim n-oo
,
-
r,
d(An)
exist for some infinite path (1.1.1), then the function p: r + R+ is harmonic. The next result shows that in this way we can obtain harmonic functions associated with every ergodic central measure.
THEOREM Let. M be an ergodic central measure on the space T. Then for M-almost all paths t = (Ao, XI,. . . ,A,, . . . ) the limits d(A An) p(X) lim d(Xn)
n+m
exist for all X E I' The proof of this theorem is an adaptation of the standard proof of Birkhoff's pointwise ergodic theorem (see, for example, [46],Appendix 3). We begin with the analysis of the maximal ergodic theorem.
$1. ERGODIC METHOD
61
Let us denote by Xm(t) the vertex Am of the path (1.1.1) a t the mth level Fm, and let am(t)be the initial interval of length m of the path t. Given a measurable function g : T + R and a path t E T , denote by
the mean value of g on the finite subset of paths <,(t) that coincide with t from the mth level. Consider now the set E = E ( g ) of paths t E T such that
(1.1.7)
s m ( g , t ) > 0 for at least one m = 1 , 2 , . . .
Clearly, the set E is measurable. LEMMA1 . The inequality
holds for every central measure M and every measurable function g : T
-t
R.
<
PROOFOF THE LEMMA.Let m n and p E r,. Denote by E n ( p ) the set of paths t E T such that ( a ) Xm(t) = p and sm ( g ,t ) > 0 , and ( b ) s j ( g , t ) 5 0 for m < j n. The set E,(p) consists of entire classes of the partition 5,. Denote by T the quotient space with respect to the measurable partition <,, and let M be the quotient measure of M on T . Since M is a central measure, we have
<
If En is the set of paths from E such that sm(g, t ) > 0 for some m the union of disjoint sets
< n, then En is
and
Since the sets E n increase and E = Ur==,E n , we obtain
g ( t ) d M ( t ) = lim n-03
The proof is complete. LEMMA2. Let M be a central measure, and let f : I', -+ R be a function defined on some level of the graph I'. Then for almost all paths t = ( X I , . . . , An, . . . ) E T there exists the limit
62
1.
BOUNDARIES AND DIMENSION GROUPS O F CERTAIN GRAPHS
PROOF.Given arbitrary rationals a < b, set
Let us check that M(Ea,b)= 0. Applying Lemma 1 to the function
yields
Indeed, if A(.9) = {t E T
-
supsn(f(Xm(t)),t)> b}, n>l
>
then A(g) E . Conversely, if t @ E, then sn(f(Xm(t)),t) 0 and A(g) that A(g) = E . Analogously, one can obtain the inequality
c E, so
Thus bM (E) 5 a M ( E ) , whence M ( E ) = 0, and the lemma follows. One can verify that f (t) = f (t) for almost all t E T . Applying Lemma 2 to the function f ( p ) = Sx, / d(X) completes the proof of the theorem.
$2. Combinatorial examples of branching graphs The central problem in the study of a locally semisimple algebra is the computation of its dimension group, that is, the Grothendieck group with a natural ordering structure. Another important problem is the description of finite and semifinite traces. As shown in Section 1.7 of the Introduction, each branching graph corresponds to a certain LS-algebra. The problem of describing the traces of this algebra and its dimension group can be restated in terms of the original graph, and may have applications beyond the theory of LS-algebras (see, e.g., [130]). Here we will consider several examples of branching graphs of combinatorial origin. Combinatorial objects are identified with the paths of the graph, the number of objects of a given type X equals dim X = d(X), and the edges of the graph, together with their (integral) multiplicities, express typical recurrence relations for these objects. 2.1.
Markovian families of classifications. If the multiplicity function
K(X, A) takes integral values, then it is convenient to regard the branching scheme
(I?, K) as a directed graph with multiple edges. A sequence of adjacent edges starting at 8 E rois called a path; let Tnbe the set of paths of length n. Denote the initial interval s E TnP1 of a path t E Tn by ~ , ( t ) we ; obtain a projective family
whose limit is the space T = bmTn of infinite paths.
52. COMBINATORIAL EXAMPLES OF BRANCHING GRAPHS
63
Let Jn be the partition of T, into classes of paths with a common endpoint. Given s E T,-l and A E J,, denote by %(s,A) the number of elements of the set {t E Al~,(t) = s). It is clear that
An arbitrary family of finite sets (1.2.1), together with partitions En defined on these sets and satisfying (1.2.2),will be called a Markovian family of classifications. Then there exists a branching scheme (i.e., a graph together with a multiplicity function) ( r , K) that generates this family according to the above description. The vertices of the nth level r, coincide with the equivalence classes of J,, and the multiplicity of an edge K(X, A) equals %(s,A) for every s E A. EXAMPLE1. Let Tn = 23, be the set of partitions of the interval P, = {1,2,.. . , n ) of N,and let T,: T, + TnP1 be the projection that maps a partition t E Tn to its restriction t o the subset P,-l c P,. Associate with a partition t E T, the Young diagram X = X(t) whose row lengths coincide with the block sizes of t. Let us classify partitions according to block sizes, i.e., assume that tl "cn t2 if X(tl) = X(t2). Then {(Bn,Jn)) is a Markovian family. The corresponding graph coincides with the Young graph y , and the multiplicity function equals
where
A = (lP1,. . . , kPk+l, (k + l)Pk+l-l ,... ), + l ) P k + l , . . . ).
A = ( I P 1 , . . , kPk,(k
The branching scheme ( y ,x B ) will be called the Bell scheme (Figure 14). Infinite paths of the Bell scheme are identified with partitions of N.
FIGURE14. Branching of partitions of a set.
64
1.
BOUNDARIES AND DIMENSION GROUPS OF CERTAIN GRAPHS
EXAMPLE2. In Example 1, consider the coarser classification of partitions according to the number of blocks (without regard to their sizes). This family is Markovian. The corresponding branching scheme will be called the Stirling scheme of the second kind (see Figure 17 in $4, where pk = k 1, vk = 0). The Stirling numbers of the second kind coincide with the dimensions dim $9 of the vertices of r.
+
EXAMPLE3. Let T, = 6, be the set of permutations of the set P,. Let T,: T, + Tn-1 be the projection that maps a permutation to its derived permutation on the subset Pn-1 c P,, and let be the partition of 6, into conjugacy classes. Then { ( e n ,<,)I is a Markovian family. Its branching graph coincides with the Young graph, and the multiplicity function equals
<,
The branching scheme ( y ,xc) will be called the scheme of conjugacy classes (of symmetric groups), see Figure 15. Infinite paths of this scheme can be identified with partitions of N whose elements are linearly ordered (as a finite or infinite interval of Z). The dimension dimA in this scheme coincides with the size of the conjugacy class A c 6,.
FIGURE 15. Branching of conjugacy classes of 6,. EXAMPLE 4. In Example 3, consider the coarser classification of permutations according to the number of cycles. This family is also Markovian. The corresponding branching scheme is the Stirling scheme of the first kind, see $4 below. EXAMPLE5 . Let T, be the set of tabloids (see [26])with elements from P,, and let the projection T,: T, + Tn-1 delete the element n E P,. Let be the classification of tabloids up to Young diagram. We again obtain a Markovian family;
<,
$ 2 . COMBINATORIAL EXAMPLES OF BRANCHING GRAPHS
65
its scheme (the Kingman scheme (see [ 1 2 9 ] ) )is shown in Figure 16. The branching graph coincides with y, and the multiplicity function equals
The above definition means that the scheme ( y ,x K ) is the branching scheme of the representations of the symmetric groups induced from the identity representation of the Schur subgroups ex, x . . . x ex, c 6,. It is interesting that this scheme appeared in [130] in connection with a problem of populational genetics.
FIGURE16. Branching of induced representations of 6,. EXAMPLE 6. An attempt to classify tabloids according to the number of rows leads, in contrast to Examples 2 and 4, to a non-Markovian family. For example, for the twc-row tabloids sl = (I$) and ~2 = and the class A of two-row tabloids from T5,we have 2(sl,A) = 2 # 1 = k ( s 2 , A ) .
(t:)
2.2. Similarity of branching schemes.
DEFINITION.Branching schemes (r,x l ) , (I', x 2 ) with a common branching graph r are said to be similar if there exists a positive function g on the vertex set of r such that
for all edges (A, A ) .
1.
66
BOUNDARIES AND DIMENSION GROUPS O F CERTAIN GRAPHS
EXAMPLE 7. The Bell branching scheme (Example 1)is similar to the Kingman scheme:
where gB (IP1, 2P2,. . . )
= p1!p2!. . . .
EXAMPLE 8. The scheme of conjugacy classes (Example 3) is also similar to the Kingman scheme: (1.2.8) where gc(lP1, 2P2,. . . )
xc(X,A) = S C ( X ) ~ K (A)gzl(A), X, =nr=l
pk!/[(k - l)!]pk.
EXAMPLE 9. If schemes (I?, XI), ( r , xZ)are similar, and J is an ideal of the branching graph I?, then the schemes (J,xl), (J,x2) are also similar. The properties of similar schemes are very alike.
PROPOSITION 1. If 91 E ex71 is a harmonic function of the scheme then the function
belongs to ex% for the branching scheme
(r,x l ) ,
(r,x 2 ) .
PROOF.We have
According t o Example 9, Proposition 1 can be extended to harmonic functions on ideals of similar branching schemes, or, in other terms, to semifinite harmonic functions. Analogously, one can check
PROPOSITION 2. If fl is a dimension function of the branching scheme ( r , x l ) , then
is a dimension function of the similar scheme
(r,x2).I n particular,
Let R(Q) = R @z Q be the rational hull of the dimension group R, and let R+(Q) be the cone over Q generated by R+.
COROLLARY. The rational hulls of the dimension groups of similar branching schemes are isomorphic as ordered groups. Propositions 1, 2, and the corollary allow us to carry over the results obtained in 93 below to the graph of conjugacy classes and the Bell graph. Note that the principal object of the paper 11291 is a family of probability distributions {cp(X) dimX), X E y,, n = 1 , 2 , . . . ( a partition structure). For similar branching schemes these distributions coincide.
53. T H E BOUNDARY AND DIMENSION GROUP O F T H E KINGMAN GRAPH
67
53. The boundary and dimension group of the Kingman graph
Let us describe the boundary and dimension group of the branching scheme
(y,xK)from Example 5. The answers and techniques are close to the corresponding results for the Young graph ([17],Chapter 3, 57). The description of the boundary was first obtained by Kingman [129]. Let mx, X E Y, be the monomial symmetric functions. Their extended versions are defined by P1 y k
(1.3.1)
M ~ ( a 1 , ~. .2. ,; Y ) = k=O
--m(1~~-*,2~2 ,... )(al,a:!,...), Ic!
where X = ( l P 1 , 2Pz,. . . ) . THEOREM1 [129]. The boundary & of the Kingman scheme can be identified with the space of sequences a = ( a l , a 2 , . . . ) such that
If 7 = 1 -
a~li,then indecomposable harmonic functions are of the form
PROOF. The functions MA satisfy the identity
Denote by A the ring generated by the functions MA,A E y . According to [17], Chapter 2, 55, the points of the boundary can be identified with ring homomorphisms cp: A + R such that cp(MA) 0 for A E Y. The values (1.3.3) at the points of & determine exactly such homomorphisms. Completeness of this description of the boundary can be proved by the ergodic method ([17], Chapter 1, 59). Let dim(X, A) be the number of paths, in the graph ( y , w K ) ,connecting vertices A, A. It follows from Birkhoff's ergodic theorem that each function cp E e x 7 can be obtained as the limit
>
for an appropriate sequence A ( ~ E) YN, N = 1 , 2 , . . . . Consider the factorial monomial functions
where (x)k = x(x - 1) . . . (x - Ic+ 1)are factorial powers, and the sum is over various factorial monomials. It is easy to check that for the scheme (y,xK) dim(X,A) - 1 - -mx(A1,A2,. dimA (N),
. .),
where A = (A1,A2,.. . ) E YN, and n = IXI. If the Young diagrams A ( ~ grow ) so that the limits of the relative row lengths llLN)
lim -= a k , N
N--too
Ic=1,2 , . . . ,
68
1.
BOUNDARIES AND DIMENSION GROUPS OF CERTAIN GRAPHS
exist, then lim -mx(Al(N),
( N ) , A$N)
,...
N+,
Since the limits (1.3.8), (1.3.9) exist or not simultaneously, the theorem follows. Formulas (1.3.8) interpret the parameters a l la 2 , . . . as the frequencies of rows of the growing Young diagram. Let us describe the supports of the functions cp,. Let Ikbe an infinite diagram with k infinite rows, and let I; be the union of Ikwith one infinite column, k = 1 , 2 , .. . , cc (so that I, = I;).
xai
PROPOSITION 2. Let = 1, a k # 0, and a k + l = 0. Then cpx(A) > 0 if and only if A c Ik. If, o n the contrary, y = 1 - C ai > 0, then cpx(A) > 0 exactly when A c I,. PROOF. This follows from (1.3.1). Denote by {hx), X E Y, the basis of complete homogeneous symmetric functions. Let HAbe the differential operator, in the ring A of symmetric functions, conjugate t o the multiplication by hx (see [95]).
THEOREM 3. Semifinite indecomposable harmonic functions on the Kingman graph are of the form
on the ideal of Young diagrams X such that
Condition (1.3.11) determines the ideal of finiteness of functions cpx(A) > 0. The number of nonzero frequencies al, a 2 , . . . is assumed to be finite. The proof is similar to that of Proposition 4.4 from [16], see also [126]. Define the jet of a polynomial Q E A as the family {H,,Q), X E y . We say that the jet is positive on a subset E c E if the conditions (H,Q)(a; y) = 0 for all (a;y) E El
7
c u,
T
# u,
and (HuQ)(a;7) $ 0
on E
imply ( H u Q ) ( a ;7) > 0 for Let Ek ={a E Elak
EL
=
{a E Elak
(a;y ) E E.
> 0, a k + l = O , y = 0 ) and
> 0, a k + l
= 0,
y
> 0),
k
=
1 , 2 , .. . , cc (Em = EL).
The general positivity theorem ([17], Theorem 2) and Theorems 1 and 3 imply
54. STIRLING TRIANGLES
69
THEOREM 4. The following conditions on Q E A are equivalent: 1) There exists a number n such that all coeficients of the expansion
are nonnegative. 2) The jet of Q is positive on all subsets Ek, Ef, for k = 1 , 2 , .. . , ca.
EXAMPLE. The function Q = (M(2)- M(12))2does not satisfy the conditions of the theorem, though all finite and semifinite traces are nonnegative on Q. Indeed, if a = (1/2,1/4,1/8,. . . ) E Em, then Q ( a ) = 0. The dimension group R of the Kingman graph is a ring which can be identified with the quotient ring A/(M(l) - 1) with respect to the ideal generated by the relation y C ai = 1. Theorem 4 describes the cone in the dimension group.
+
54. Stirliiig triangles
Let us now consider examples of branching schemes (I?, x ) , where I? is the Pascal triangle and the edge multiplicities are determined by sequences p = (po,p l , . . . ) and v = (vo,vl, . . . ), Figure 17. We call them generalized Stirling triangles.
FIGURE17. Generalized Stirling triangles.
A:",
The vertex at the ith row and j t h column of the graph I? will be denoted by where n = i j is the number of the level, i , j = 0 , 1 , 2 , . . . .
+
PROPOSITION 1. For a generalized Stirling triangle, (1.4.1)
dim A/") = hn-j (po,p l , . . . , p j ; uo, vl , . . . , vnPl).
PROOF. The extended complete homogeneous symmetric functions (see [17], Chapter 3, 57) satisfy the recurrence relation (2):
70
1.
BOUNDARIES AND DIMENSION G R O U P S O F CERTAIN G R A P H S
Consider the functions
They satisfy the relations
THEOREM 2. Assume that uo = y = . . . = 0 (a generalized Stirling triangle of the second kind). Then the dimension group can be identified with the group of polynomials Z[l/x]. The boundary E is the union of the interval [p(O),+oo], where p(O) = ~ u p ~ > ~and p k the , points x = pn with pk < pn for k < n. Let P E Z[l/x], and let n be-the least value of k such that P ( p k ) # 0. Then P belongs to the cone of the dimension group if and only if
for x
> p(n) =
-
pk, and for x = pk if p j
< pk for n 5 j < k .
PROOF.The group generated by the functions p y ) ( x ) coincides with Z[l/x]. By the ring theorem ([17],Chapter 2, §5), harmonic functions from e x 7 are determined by ring homomorphisms .p: Z[l/x] R with ~ ( h ? ) ) 0. It is easy to see that such homomorphisms are of the form-
-
>
Primitive ideals of the graph 1. ([17], Chapter 1, $11) are of the form {A?)},
j
> k,
or { h y ) } ,n - j 2 k . In both cases they are isomorphic to generalized Stirling triangles, maybe with different values of the parameters po, p1, . . . . From previous considerations the values of traces on them are known, so we can apply the positivity theorem ([17], Chapter 2, §5), which yields the desired description of the cone.
- -
COROLLARY 3 (Poincark [147]). For the Pascal triangle (pk 1, vk 0, k = 0 , 1 , 2 , . . . ), the cone i n Z[l/x] consists of functions P ( x ) such that P ( x ) > 0 for x > 0.
+
COROLLARY 4. 0, k = 0 , 1 , 2 , . . .), consists of functions and P ( k ) > 0 for k
-
For the Stirling triangle of the second kind (pk = k 1, vk the boundary is E = { 1 , 2 , 3 , .. . , c o . The cone i n Z[l/x] P such that P(l)= . . . = P ( n - 1) = 0 for some n = 1 , 2 , . . . , > n.
Note that the examples from [30] related to the Pascal triangles are similar to generalized Stirling triangles of the second kind. Another interesting class of examples, generalized Stirling triangles of the first kind, arises when po = p1 = . . . = 0. Here the answers depend substantially on the rate of growth of the sequence VO, ~ 1. .,. . Let us first assume that l / V k < co. Denote by e k ( x o , x l , .. . ) the elementary symmetric functions in arguments xi = l/vi, i = 0 , 1 , 2 , . . . , and set
xpro
54. STIRLING TRIANGLES
THEOREM 5. The functions decomposable.
( ~ k for ,
71
k = 0 , 1 , 2 , . . . ,GO, are harmonic and in-
PROOF.Harmonicity follows from the relations ~ ~ ( Y o , Y I , . .= . ) e i ( ~ 1 , ~ 2. .,). + Y O ~ ~ - I ( Y I , Y Z , . . . ) .
Nonnegativity is obvious. Indecomposability follows easily from the fact that the support of the functions (1.4.6) consists of finitely many columns or one row of r.
COROLLARY 6 . Consider a q-version of the Pascal triangle with parameters pk = 0 , vk = q k , k = 0 , 1 , 2 , . . . , where q > 1. Then the functions ( ~ k k, = 0 , 1 , . . . ,co, exhaust the boundary E .
PROOF.It follows from [49], s1.3, that
where i = n - j , x = l / q . The group generated by these functions coincides with the ring Z [ x ] . Completeness of the list follows easily from the ring theorem. Note that we have a "phase transition" as q
E
-+
1: the countable boundary
= ( 1 , l / q , l / q 2 , . . . , 0 ) turns into the interval [0,11.
PROPOSITION 7. The q-Pascal triangle is similar to the Stirling triangle of the second kind with parameters pk = q k , vk s 0, k = 0 , 1 , 2 , . . . . PROOF.The function g ( A y ) ) = vjvj+l . . . vn-1 implements the similarity of a generalized Stirling triangle of the first kind and the triangle with simple vertical edges and horizontal edges of multiplicities x(Xj9,A$:')) = vj/v,. For the q-Pascal triangle this gives a "transposed Stirling triangle of the second kind. COROLLARY 8. The dimension group of the q-Pascal triangle is isomorphic to the ring Z [ x ]with the cone consisting of polynomials P ( x ) such that P(1) = P ( l / q ) = .. . = ~ ( l / q ( ~ - = l ) 0) ,
and
p ( l / q k ) > 0 for k
2n
for some n = 0 , 1 , 2 , . . . . Let us now consider the functions
on the Stirling triangle of the first kind. It is easy to see that they are nonnegative and harmonic for 0 5 x 5 oo.
CONJECTURE. If C 1/vk = co, then these functions are indecomposable and exhaust the boundary E . PROPOSITION 9. The conjecture is true if each value occurs i n the sequence vo, y,. . . with infinite multiplicity.
72
1.
BOUNDARIES AND DIMENSION GROUPS O F CERTAIN GRAPHS
PROOF.In this case, the group generated by the functions cp?'(x) admits an order-respecting multiplication, and we can apply the ring theorem. The author does not know whether the conjecture is true for the triangle from Example 4. Kingman [130] showed that the functions corresponding to cp, in the graph of conjugacy classes, the so-called Ewens distributions, are indecomposable in this graph. In the case of the q-Pascal triangle it is easy to obtain an expansion of the functions cp, in an explicit form.
PROPOSITION 10. Let II = n r = o ( l + x l q k ) and (1.4.9)
1
Pk = -
(qxIk
II ( q - l)(q2 - 1 ) . .. (qk
-
1)'
k=O,l, ... .
Then for all A?) E J?
PROOF.This follows from Euler's identity ([63], 2.2.6)
It is natural t o regard the distribution {Pk)r=oas a q-analogue of the Poisson distribution with parameter y = qx/(q - 1). We conclude this section by observing that harmonic functions on a branching graph determine central measures in the path space of this graph ([IT],Chapter 1, 58). Examples 1 and 3 lead to interesting measures in the spaces of partitions of N,see [9], [130].
CHAPTER 2
The Boundary of the Young Graph and Macdonald Polynomials In this chapter we describe the structure of the group algebra of the infinite symmetric group 6,. We find its two-sided ideals and characters (finite and semifinite). Most results are stated in general terms of branching graphs. This leads us naturally to similar questions for the two-parameter deformation of the Young graph associated with the branching of Macdonald symmetric polynomials. A considerable part of the chapter is devoted to discussion of properties of these polynomials, as well as of the more general class of generalized Hall-Littlewood symmetric polynomials introduced in [35, 1191.
$1. Ideals of the group algebra C[6,] 1.1. Branching of the characters of the symmetric groups. Let 6 , be the group of finite permutations of a countable set. In what follows we consider this group with a fixed structure of the inductive limit of finite symmetric groups 6 , = l& 6 , (none of our results on 6 , will depend on the choice of this structure). Thus we may regard 6 , as a group of permutations of the set N of positive integers, where 6 , = {a : a ( k ) = k, Ic 2 n). Denote by @ [B,] the complex group algebra of B,, i.e., the algebra of finite formal linear combinations of permutations from 6,; the completion of @ [B,] is the group C*-algebra C*(6,). Since @ [ B , ] = l a @ [ B , 1, the algebra @ [ B , ] is locally semisimple. A Young diagram is a finite ideal X in the partially ordered set N2, i.e., a subset X c N2 that contains, along with each "square" (Ic, 1 ) E A, all squares with smaller indices: ( i ,j ) E X provided that i 5 Ic, j 1. Let y, be the set of Young diagrams with n squares. The set y = UrE0yn of all Young diagrams is partially ordered by inclusion; its Hasse diagram is the Young graph (Figure 1, p. 6), which is a graded graph: a grading on y is defined by the number of squares in a diagram. Standard constructions of the representation theory of finite groups (see [26], 1501) associate with a diagram X E y, an irreducible complex representation vXof the group 6 , with character XX. The following theorem is well-known.
<
BRANCHING THEOREM(see [26]).Irreducible characters of symmetric groups satisfy the following relation:
where A E y,+l, a E 6,. In other words, the restriction of the representation of the group 6,+i to 6 , decomposes into the direct s u m @A,xEy, vX.
vA
74
2. BOUNDARY OF THE YOUNG GRAPH AND MACDONALD POLYNOMIALS
This fact means that the partially ordered set y determines the branching scheme (or Bratteli diagram, see [74]) of the inductive family of finite-dimensional a l g e b r a s @ [ 6 1 ]c @ [ 6 2c] . . . c C [ 6 , ] c . . . . A Young tableau u with diagram X E y, is a maximal strictly increasing chain Q) = uo c ul c . . . c u, = X of the partially ordered set y , i.e., a path in the Young graph starting a t 0 and ending at A. The set of such tableaux will be denoted by T(X), and its cardinality (the number of paths leading to A) by dimX, since it coincides with the dimension of the corresponding irreducible representation v'; the dimension dim X can be computed either by the classical Frobenius formula, or by the hook-length formula (see [45]). Given A, A E y , denote by dim(X,A) the number of maximal strictly increasing chains from X to A. Iterating (2.1.1) yields
for A E Y,, a E 6,, m < n. A finite bitableau is a pair of tableaux with a common end, i.e., a finite loop in the graph y . 1.2. Infinite Young diagrams and tableaux. An infinite tableau t is a maximal strictly increasing chain 8 = to c t l c . . . c t, c . . . of the partially ordered set y. An (infinite) diagram is an infinite ideal in N2. A tableau t gives rise to a diagram [ t ] = t,. It is also convenient to define a tableau by a monotone enumeration of the squares of its diagram, i.e., by the function Nt : [ t ] + N determined by the formula Nt (t,\t,-1) = n; we have t, = N t l { l , . . . ,721. We endow the set T of tableaux with the topology of a Hausdorff compact totally disconnected space by taking the cylinders F, = {t E T : h,(t) = u}, where u is a Young tableau with n squares and h,(t) is the initial interval of length n of the chain t , as the basis of neighbourhoods. Thus T is the space of infinite paths in the Young graph y . Let y, be the set of all (infinite) diagrams. The map from T to y, that associates with a tableau t its support [t] induces a quasicompact (non-Hausdorff) To-topology on y,; the basis of neighbourhoods of this topology consists of the sets Ox = { I E Y , : I 3 A}, X E Y .
UrT1
PROPOSITION 1. The space of diagrams y, is naturally homeomorphic to the primitive spectrum Prim C*(6,) of the group C*-algebra of 6,. The proof follows easily from the general description of ideals in LS-algebras (see [83]). The homeomorphism maps a diagram I E y, to the closure in C [6,] of the ideal U' = l@ U i , where U i = ker VX. Each diagram Y, 3 I # N2 is either of the form I = Ik,l= {(i,j ) : i 5 k {v ( k , l ) ) , where v is a or j I 11, k 1 2 1, or of the form I = Ill = I k ,U~ nonempty finite Young diagram. Denote by' T ( I ) the subset of tableaux t with diagram [t] = I . The closure of T ( I ) in T consists of tableaux with [t] c I .
nyn3XcI
+
+
1.3. Bitableaux and crossed products. There is an important equivalence relation on the set T. Tableaux t', ttt E T are equivalent, tt ttt, if they eventually coincide. The partition into the classes of equivalent tableaux in T will be denoted + t2 + by <; it is the intersection of the decreasing sequence of partitions
52. CHARACTERS OF 6,
75
. . . , where En is the partition into the classes of tableaux coinciding from the nth position. A pair of equivalent tableaux (t', t"), t' t", will be called a bitableau. Let us endow the space B of bitableaux (i.e., the graph of the <-equivalence relation) with the topology of the inductive limit B = 1 4 B,, where Bn consists of pairs of tableaux coinciding from the n t h position (the graph of the <,-equivalence relation). Thus B is a Hausdorff totally disconnected locally compact space, and the diagonal Bo = {(t, t) : t E T) is homeomorphic to T . Let C(T, <) be the space of complex-valued locally constant compactly supported functions on bitableaux. The following operations of multiplication and conjugation (analogous to the corresponding matrix operations): N
turn C(T, <) into a *-algebra. The C*-completion C*(T, <) of this algebra is called the C*-crossed product constructed from the partition on T .
<
PROPOSITION 2. The algebras @ [B, ] and C(T, [) are isomorphic. PROOF. Let U ' , c C ( T , < ) consist of functions f : B + @, vanishing outside B, such that the value f (t', tr') depends only on the initial intervals hn(tl), h,(tU) of tableaux t', t". Then U ' , is a *-subalgebra, and C(T, E) = 1 4 2,. It is clear from construction that the branching scheme of the inductive family U1 c . . . c U, c . . . coincides with the Young graph y. By Bratteli's theorem [74] there exists an isomorphism i : C*(B,) + C * ( T , < ) that maps @ [ G n ]to 2,; in particular, i e [ e, 1 = C(T, I). This isomorphism maps the commutative subalgebra 9Jl of functions supported . by the diagonal Bo to the Gelfand-Tsetlin algebra of the inductive family {Bn)r==, Note that the structure of crossed product in C*(B,) depends already on the choice of an inductive family in G,. 52. Characters of the infinite symmetric group 2.1. The theorem on approximation of characters. The fundamental fact, which is very useful for computing the characters of a locally finite group, is that these characters can be approximated by irreducible characters of finite subgroups. We give this assertion in its strongest form.
THEOREM 1. For each normalized character cp of the group 6, there exists a tableau t E T such that
xtn( a ) cp(a) = lim n+w dim tn for all a E 6,. In other words, each character of 6, is the limit of a certain sequence of normalized irreducible characters of the groups 6, as n + m. This theorem is a direct consequence of the theorem from 51 of Chapter 1 and the correspondence between central measures and characters established in Section 1.4 of the Introduction.
2. BOUNDARY O F T H E YOUNG GRAPH AND MACDONALD POLYNOMIALS
76
This method for computing ergodic measures was suggested in [lo]; it was used for constructing measures that are invariant with respect t o the groups 6,, U(co), etc. COROLLARY 1 . Let M be a locally finite central ergodic measure on T, and 0 < cp(X) < co. Then, for almost all tableaux t E T, lim
n+m
for all X with cp(X)
dim(A, tn) dim(,!, tn)
-
y(X)
# 0.
2.2. Asymptotics of the number of skew Young tableaux. It follows from Theorem 1that in order to find all characters of 6, it suffices to find the limits of various sequences of irreducible characters of the groups 6, as n + co. In order to describe explicitly the conditions under which the limits of irreducible characters exist, it is convenient to use the following version of the F'robenius coordinates of a Young diagram X E y : fk
(2.2.3)
+i, : (k, i ) E A) - k + i,
(A) = max{i : (i, k) E A) - k
gk (A) = max{i
where k = 1,2, . . . ,r(X); r(X) = max{k : (k, k) E A) is the length of the main diagonal of A*). Given a pair of sequences f = (fl, f 2 , . . . ), g = (gl, g2 . . . ) (finite or infinite), let
LEMMA1 . Let a, E 6, be a pennutation with a single nontrivial cycle of length m. There exists a polynomial P, such that xX(orn) - Prn(X) + pm(Pl(X),. . . ,Pm-l(X)) dim X (n)m
--
where pm(A) = p, (f (A), g(X)), X E y n , n in f k , gk, k = 1,2, . . . , is at most m - 1.
> m.
,
The degree of P,
as a polynomial
The proof of this lemma may be found in the book [50](Chapter 6, 51, Section 5); it is not difficult to compute the polynomials Pmexplicitly. The lemma also holds for arbitrary permutations a E 6,; for example, for a = (1,2) (3,4), X" (am) dim X
-
~PT
P; (A) - 4 ~(A) 3 + (A) - 3Pl (A) n ( n - 1) (n - 2) (n - 3)
Relations of this kind will be considered separately. *)1n Figure 3, p. 21, r(X) = 5,
f = ( l l ; , 9;, 7;,
3;, 2;);
g = (7+, 4:,
3;, I;,
;).
52. CHARACTERS O F Bm
COROLLARY 2 . The following asymptotic formula holds:
where the error term tends to zero as n 4 ca uniformly i n X E Yn. The proof follows from the homogeneity of p, and the estimates
LEMMA2. The following conditions o n a tableau t E T are equivalent: 1 ) For each m > 2 there exists the limit lim
n-m
xtn
"(a,)
dimt,
- Pm
2 ) For each k = 1 , 2 , .. . there exist the limits fk(tn) lim =ak,
n-m
n
lim gk(tn) n = Pk.
n-oo
If these conditions are satisfied, then
PROOF. If the limits (2.2.8) exist, then by Corollary 2
pm = lim
n+m
Conversely, the sequences recovered from the sums p,:
xtn " m ) dimt,
> a2
a1
=pm(al,..
. :PI).
2 . . . and pl 2 jj2
> .. .
can be uniquely
if
and
then
Hence if for some k the sequence fk ( t n ) / nhas more than one limit point, then the limits (2.2.7) do not exist either. Thus the numbers a k and Pk have a clear meaning: they are the frequencies of the squares of the kth row and kth column, respectively, in the growing tableau. As follows from the main theorem, the character is determined only by these frequencies.
78
2. BOUNDARY O F THE YOUNG GRAPH AND MACDONALD POLYNOMIALS
2.3. Parametrization of characters.
THEOREM 2 (Thoma [161]). All normalized (indecomposable) characters of the group B, are given by the formula
where a1 > a 2 . . . L 0, PI 2 P 2 > . . . > 0, Cai+CPi < 1, and p,(a) is the number of cycles of length m i n a permutation a . I n particular, if a, is a permutation with a single nontrivial cycle of length m, then cp,,p(am) = p,(a,P). Thoma proved this theorem by using, first, a simple fact on the multiplicativity of indecomposable characters of B, with respect to cycles, which reduces the problem to computing the values of the character on one-cycle permutations; and second, the fact that the generating function for these values satisfies a certain functional equation. Thoma solved this equation using a deep theorem of Nevanlinna's [144] from the theory of integral functions. The meaning of the parameters a, which index the solutions of this equation has no explanation within this argument. Our proof is based on the ergodic method and passing to the limit from finite groups.
PROOFO F THE T H E O R E M . Let cp be a normalized character. By Theorem 1 there exists a tableau t E T such that ~ ( o = ) lim for all o E 6,. By Lemma 2 the limits (2.2.8) exist, and cp(a,) = pm(al, . . . ; PI, . . . ). Finally, according to the above-mentioned multiplicativity, the value of the character on an arbitrary element a E B, can be expressed as the product of its values on one-cycle permutations: cp(a) = - cp(a,)~m(~).
nm,2
Note that the generalization of (2.2.5) to arbitrary permutations yields (2.2.9) directly for an arbitrary element a E B,, without using the multiplicativity. Thoma's multiplicativity theorem, as well as its generalization to the group U ( m ) due to Voiculescu, is a consequence of the following general theorem: A n R-valued indecomposable positive functional on a Riesz ring is a ring homomorphism. The dimension group of the algebra @[B,] is a Riesz ring (see [126]). Denote by M,,p the central ergodic measure associated with the character cp..,p by formula (6). p, ( a , P ) P m , where p is a Young diagram with The functions pp(a,P) = ) given by (2.2.4), are called the extended row lengths pl,p2,. . . , and p m ( a T ~ is Newton functions. Since all symmetric functions can be expressed in terms of Newton functions by well-known formulas, they give rise to the corresponding extended functions in a, p. In particular, the extended Schur functions s x ( a ,P) are defined by Frobenius' formula
n,,
pP(a,B ) =
xA(p)
0).
A
Comparing this formula with (6) yields
COROLLARY 3. The values of the central ergodic measure Ma,8 on the cylinders F,, u E T(X), are given by the extended Schur functions: M..,~(X)= s x ( a ,P).
52. CHARACTERS OF 6,
lim
79
COROLLARY 4. For almost all, with respect to Ma,p, tableaux t E T the limits = a k and lim = pk exist for all k = 1 , 2 , . . . .
The most surprising consequence of Theorem 2 is that the zero frequencies a = p = 0 correspond to the unique character S,, the character of the regular representation. The corresponding central measure, called the Plancherel measure, will be studied in detail in Chapters 3 and 4. It turns out that the rows and columns of almost all tableaux with respect to this measure grow as fi,and after an appropriate scaling (1/fi) the shape of the tableau becomes stable. Thus, to obtain all limiting characters, it suffices to consider only tableaux with row and column lengths either linear in n, or of the specific order fi. The parameters a, ,D have another interpretation related t o the RobinsonShensted-Knuth (RSK) correspondence and the construction of the representation of type 111 associated with the character p,,p (see [14]). In this connection, we mention here only the following fact. If p is a Bernoulli measure in the space of sequences with a common distribution of coordinates a = (al,a 2 , . . . ), CEl ai = 1, then the value p a I o ( a )is the p-measure of the set of fixed points with respect to the permutation a of coordinates:
This interpretation is related to another proof of Thoma's theorem suggested by Vershik and the author in [127]. Namely, there exists a map from the space of IR into the space of tableaux T (the generalized RSK-transform) sequences X = such that each central ergodic probability measure on T is the image of a product measure. LetX~Y~beaYoungdiagram a n, d f i = fi(X)+;,iji = g i ( ~ ) + ; , i = 1, . . . , r . Set 6f = Bf1 x Bj2 x . . . , 6 9 = Bg1 x Bg, x . . . , and let 4' be the character of en+,induced from the Young subgroup 6f x 6 9 by the character 1x sign, where 1is the identity character, and sign is the signature character. It is not difficult t o check that if the frequencies (2.2.8) exist for a sequence of diagrams {t,), then
ny
Gtn (a) lim Gtn ( E ) = cpa,p(u)
n+,
for all a E 6, (here E is the identity permutation). Thus Theorem 2 implies COROLLARY 5. Allfinite characters of 6, are the limits of induced characters. The direct (independent of Theorem 2) proof of this corollary uses the abovementioned generalized RSK-correspondence; it has been published in [127]. It follows from (2.2.11) that for a large diagram X the dominant part (in the sense of relative dimension) of the decomposition of the induced representation with character 4x consists of irreducible representations with characters close to X A . This statement may be called the law of large numbers for induced representations. It seems to be of a quite general character. 2.4. Semifinite characters of 6,. In this section we compute all infinite characters of the group 6,. This allows us to describe the cone of positive elements in the KO-functor of this group.
2. BOUNDARY O F T H E YOUNG GRAPH AND MACDONALD POLYNOMIALS
80
Given a tableau t E T(Iky,'), denote by vt : dk>') -+ W the restriction of Nt to v("') = v (k, I)*). Let f E T(Ik,') be a maximal strictly increasing subsequence in the sequence of diagrams {t, n Ik,L)p=l. The tableau t can be recovered from f and vt, so that the whole space of tableaux on Iky,' is partitioned into the union U T, of disjoint parts T, = {t : vt = v), where v : u ( ~ > '+ ) W is a monotone embedding, each T, being embedded in T(Ik,'). Given a finite collection of numbers a1 . . . a k > 0, 2 ... > 0, ~Zk_,a(CfzlPi= 1, let Ma,p be the corresponding central ergodic probability measure on T(Ik,'). Define a measure M:,p on T(IL 1) as follows: on each T, it is the inverse image of M,,B under the embedding t + f. In other words, if u is a Young tableau of shape X and v(k)l)c A, then by definition
+
>
>
+
>
If u, w are Young tableaux of shape X and c A, then u, ZZ E T(XnIk,1)and Ma,p(F,) = M a , ~ ( F a )so , that the measures M:,p are central. Their ergodicity is also obvious, since any Young tableaux u E T(X1), w E T(X2) can be extended t o tableaux with a common diagram X = X1 U X2. It is not difficult to check that the ) [ t] c k Iy l . poles of the measure M,",p are precisely the tableaux t with u ( ~ , ' @ We will show that each infinite central ergodic locally finite measure is pioportional to one of the measures M&. ~
(
~
1
'
)
LEMMA3. Let M be a central ergodie locally finite or finite measure. Denote by IME yoothe union of all diagrams X c N2 with M ( X ) # 0. Then M(A) # 0 for all A c I M .
-
PROOF. If M ( X ~ ) , M(X2) # 0, then M(X1 U X2) # 0; otherwise {t : [ t ] > X I ) and {t : [ t ] 3 X2) are disjoint J-invariant sets of positive measure.
LEMMA4. Let {t,} be a sequence of diagrams such that the limiting frequencies Pi = 1. Then m, = lvnl = o(n) and (2.2.8) exist and a k , P1 > 0, C ; ai
+
(2.2.13) as n
+
dim(& t,)
-
nmn
-dim vn dim p,
m,!
co, where p, = (tn\X) n 4,1, v,
=
(t,\X)
\ Ik,l, A E Y
(Figure
4,
p. 23).
PROOF. AS described a t the beginning of this section, we can associate with each tableau w on t,\X a p,-tableau u and a monotone embedding v : v, 4 { I , .. . , n) which uniquely determine w. Thus dim(X, t,) does not exceed the total number C,"" dim v, dim p, of all such pairs (u, v). Pick a subsequence r, -+ m such that m,/r, 4 0, r,/n 4 0 as n + co. Let T, be the set of p,-tableaux u m,, g1 (urn) m,; it follows from (2.2.8) that lim ITn(/ dim p, = 1. with fk(ur,) Finally, if u E T, and v : v, + {r, 1 , . . . ,n ) is a monotone embedding, then the pair (u, v) corresponds to a certain (t,\X)-tableau w, so that dim(/\,t,) 2 dim v, . IT, 1. Both estimates on dim(X, t,) are equivalent as n + co to the right-hand side of (2.2.13).
>
he sum is taken in Z2.
>
+
52. CHARACTERS OF Bm
81
THEOREM3. Each locally finite ergodic central measure M o n the space of tableaux T is proportional to one of the measures M&. PROOF. Let X be a Young diagram with 0 < M(X) < co. By Corollary 1 there exists a tableau t E T such that (2.2.2) holds. Dropping down to a subsequence, if necessary, we may assume that for the sequence of Young diagrams {t,)p?lthe limiting frequencies a = ( a l , a 2 , . . . ), ,B = (PI,B2, . . . ) in (2.2.8) exist; let m be the corresponding probability measure. Then lim
dim(A, t,) dim t ,
= m(A)
AEY.
for all
If m(X) # 0, then dim(A, t,) dim t, m(h) +dimt, dim(X, t,) rn(X) ' so that M(A) = am(^) and M is a finite measure. Thus the case I, = N2 is not interesting, and we may assume that I, = Ik,'. Suppose that m(X) = 0, i.e., that X\Ik,' # 0. Then v = X-('">')is a nonempty Young diagram*),and by Lemma 5 we have
for every Young diagram A with A-("') = v. Comparing with (2.2.2), we see that the measures M and M& are proportional. Going over from central measures on T to characters of 6, by general formulas, we obtain the main result of this section.
>
>
THEOREM4. Let a1 _> . . . a k > 0, 2 . . . PL> 0, and let v be a nonempty Young diagram. The formula
C: ai +
~i
=
1,
c X c I;,",' determines an indecomposable semifinite character of the group em**). If x-(~%')c v , then pE,,p(EA)= co; for X $ I;,, we have pL,p(Ex) = 0.
for
~
(
~
1
'
)
#
Each (infinite) semifinite character of 6, the characters p i , p .
coincides, up to a factor, with one of
Note that the character p&, v E Ym,can be described as the character induced from the subgroup 6, x 6, of permutations a E 6, with u(k) < m for Ic < m:
*)M-(~= - ~N2 ) ( p - ( k , l ) ) , where the subtraction is in 2 '. * * ) ~ e c a lthat l sA(a, P ) is the extended Schur function.
82
2. BOUNDARY OF THE YOUNG GRAPH AND MACDONALD POLYNOMIALS
$3. Generalized Macdonald polynomials and orthogonal polynomials 3.1. The Hall-Littlewood polynomials were originally introduced by P. Hall [I041 and D. E. Littlewood [137]. An excellent exposition of their properties and numerous applications to the theory of (ordinary, projective, modular) representations of symmetric groups and full linear groups can be found in [49]; see also [143]. In a recent paper [139], Macdonald introduced a more general family of symmetric polynomials and described their remarkable properties. As a limiting case this family contains Jack polynomials, in particular, zonal spherical functions of the symmetric spaces GL(n, R)/O(n) (cf. [112]). Combinatorial properties of Jack polynomials were also studied by Stanley [159] and Hanlon [106]. The main purpose of this section is to characterize the HL-functions introduced by Macdonald within a much wider class of polynomials which depends on infinitely many parameters. In a certain sense, the latter are infinite-dimensional versions of the generalized Legendre polynomials defined by Fejkr [go]. From this viewpoint Macdonald HL-functions correspond t o continuous q-ultraspherical polynomials, first considered by Rogers [154] in his famous paper on the Rogers-Ramanujan identities. 3.2. Let us describe our approach in more detail. In this section we borrow the notation and terminology concerning partitions and symmetric functions from 1491. Let h = @ A, = A 8 @ be the graded algebra of symmetric polynomials with complex coefficients in variables x = (xl, x2, . . . ), and let P = U P, be the set of partitions of positive integers. Given X E P, denote by px the corresponding monomial symmetric function, and by px the corresponding Newton power sum. Consider the family of symmetric bilinear forms (., .), on A, depending on a sequence of complex parameters w = (wl, w2,. . . ), which is defined by the formula
where ZX =
l m l m l !2m2m2!.. .
and
WX
=
wX2
for a partition X = (lm12mZ. . . ) with multiplicities m l , ma, . . . . We fix an arbitrary total ordering 5 on P, that agrees with the natural (dominance) ordering, for example, the reverse lexicographic ordering. Define the generalized Hall-Littlewood polynomials (GHL-polynomials, for short) Px(x;w) as the result of the orthogonalization process applied t o the basis {mx(x)) with respect to the form (., .) and the chosen ordering. Thus Px(x; w) = mx(x)
(PA, P,), Denote by Qx(x;w)
= bx(w)Px(x;w)
+
uxP(w)mp(x), P
the adjoint GHL-functions, defined by
$3. GENERALIZED MACDONALD POLYNOMIALS
3.3. Several examples are worth special mention. (S) Schur case: 1 n=1,2,....
Here Px(x; w) = Qx(x; w) = sx(x) are the Schur functions. (H) Hall-Littlewood case:
where Px(x; w) (J) Jack case:
= Px(x; t) are the classical Hall-Littlewood n = l , 2,...,
w,=a,
(J,)
polynomials.
where Px(x;w) = h x ( a ) J x ( x ;a ) coincide, up to the factor hx(cu) independent of x, with the Jack polynomials. Generalized Jack case, m 2:
>
w h =a;
=
W,
1 if
n
# 0 mod
m.
As far as I know, this case was not earlier considered in this context. For m = 1 we arrive at the ordinary Jack case (J). (M) Macdonald case:
This case was studied in 11391. All previous examples are special or limiting cases of this one. For instance, in order to obtain the case (J,), one should take q = r + a . E, t = r E, where r is the mth primitive root of unity, and let E -, 0.
+
3.4. We present a list of basic properties of the HL-polynomials which were proved by Macdonald in the case (M). By continuity these properties are also valid in the Jack cases (J) and (J,). (i) Duality: w, PA(%;w) = Qxf(x; l / w ) for the algebra endomorphism w, : A + A defined by w,(p,) = (-l),+' . w, . p,, n = 1,2, . . . . Here A' is the conjugate partition of A and l l w = ( 1 1 ~ 11, 1 ~ 2 ,. .. ). (ii) ~ u ~ e r o r t h o ~ o n a l i t yLet * ) : A = (A1,. . . , A,) be a partition of length n and x = ( x l , . . . ,x,,O,O,. . .); then
Px(x; w)
= 21x2.. .x,
where A, = (A1 - 1 , . . . , A, (iii) Supertriangularity:
-
. P A * (x; w),
1).
uxP(w) # 0
implies p << A,
where << is the dominance ordering (by definition, the matrix (ux,(w)) is triangular only with respect to the total ordering 5 ) .
he meaning of this term will
be clear from 57
84
2. BOUNDARY OF THE YOUNG GRAPH AND MACDONALD POLYNOMIALS
(iv) Young-type branching: Let us define the branching multzplicities m x ~ ( w ) by P(I)(x;W) . PA(x;W) = PA(x; w);
X~AA(W) A
then mxA(w)= 0 unless A immediately succeeds X in the Young graph (i.e., A > X and IA\XI = 1, where X and A are regarded as Young diagrams rather than partitions). In this work we are most interested in the following question: to what extent do these properties force the parameter sequence w to assume one of the values indicated in case (M) and the other above-mentioned cases. 3.5. First of all, let us rewrite the duality formula (i) in an appropriate form. Define the conjugate ordering 5' on P, by
and denote by prove
Pi
the GHL-polynomials associated with this ordering. We will
THEOREM 1. For all w = (wl, wa, . . .),
It is sufficiently clear that properties (ii)-(iv) imply much more severe restrictions on w. Our next main result is
THEOREM 2. Assume that (ii) holds for (wl, w2,. . . ) with n = 2. Then the sequence W(C) = ( c . W1, c2 . wa, . . . ,ck . W b . . ), for an appropriate constant c, belongs to one of the cases ( M ) , (J), or (J,), above. Note that the "gauge transformations" w -+ w(') leave GHL-polynomials unchanged: PA(x;w")) = PA(x;w). We will prove Theorem 2 by reducing it to a remarkable theorem of Feldheim [60] and Lanzewizky [48] which characterizes the orthogonal generalized Legendre polynomials as continuous q-ultraspherical polynomials originally studied by Rogers [154]. We give a complete proof of this result, since the limiting cases (J,) were overlooked in [60], [48]. The orthogonal polynomials corresponding to these cases were studied in [65] under the name of "sieved ultraspherical polynomials of the first kind". See [67], [70] for the properties of Rogers' polynomials. At present I do not know the exact restrictions on the parameters (wl, w2, . . . ) implied by properties (iii) or (iv). $4. Orthogonalization of characters of the symmetric group 4.1. Let us introduce some notation. Denote by F, the complex vector space of central functions on the symmetric group 6,. Irreducible characters XX indexed by partitions X E P, form a linear basis in F,. Denote by cP the characteristic function of the conjugacy class with cycle structure p E P,, and let CP = zp . CP for zp = lml. ml! . 2m2. ma!. . . , where p = (lml, 2,2,. . .). Then C P is the density of the uniform probability distribution on the class p.
54. ORTHOGONALIZATION O F CHARACTERS O F 6,
There is a natural bilinear form on 3,:
where f p = f ( a ) are the values of f E Fn on a permutation a with cycle lengths p1, p2, . . . . For example, = (x', Cp) is the value of X' on the class p.
Xi
4.2. Given a sequence w = (wl, w2,. . . ) of complex parameters, define a more general bilinear form (., .), on Fn by
The weight function w(a) = w y l . w y 2 . . . . is multiplicative:
for a permutation a with cycle lengths pl, p2 . . . . By definition, (CP,C T W) = 6p7 Z p wp for p, r E P,.
4.3. Recall the definition of the natural (dominance) ordering partitions P,:
We say that a total ordering
< on P, p << X
<< on the set of
is comparable with the dominance ordering if implies p1
< A'.
Let X I denote the conjugate partition of A. Define the conjugate ordering 5' as follows: p < I X if and only if p' 2 XI. This ordering is also total and comparable with the domillance ordering. Denote by r = r(X) the number of X E Pn with respect to the chosen ordering. By definition, r ( ( l n ) ) = 1 and r ( ( n ) ) = p(n). Figure 18 shows the Hasse diagram of the set P6 with dominance ordering. The numbers of partitions in the reverse lexicographic ordering are denoted by figures, and in the conjugate ordering, by letters.
FIGURE 18. Various orderings on the set of partitions.
86
2 . BOUNDARY O F T H E YOUNG G R A P H AND MACDONALD POLYNOMIALS
4.4. Let Y ( l n ) , .. . , Y(") be the functions in 3" obtained by the orthogonalization of irreducible characters with respect to the form (., .), and the total ordering I.Thus
We may also write
The functions YX = YX(w)are uniquely determined by properties (2.4.3), (2.4.4) or (2.4.4), (2.4.5). 4.5. Consider the Gram matrix H = (hXp),hAP= (xX,xP),, minors. We will use partitions X and their numbers r(X). Let
and some of its
denote the principal minor of order i = r(X), and, more generally,
SO that fx = Lxx Let JxP be the cofactor of the element hpA in the determinant (2.4.6) and gx = JxxDenote by U; the determinant
j = r(p),
Clearly, for X = (1") we have f(ln)(w)=
wp/.zp and
ujl")(w) =
XFn)
= E,,
where E~ is the sign of a permutation with cycle structure p. Later in $5 we will see that for X = (n)
and u~")(w) = f(n)(w)/wp (these functions do not depend on the choice of ordering). We will use this notation to derive explicit formulas for the functions YX and transition coefficients VxP, KxP.
$4. ORTHOGONALIZATION O F CHARACTERS O F
en
4.6. Consider the functions X X defined by the formula
where
(2.4.11)
bx(w) = (YA,YX),~.
Then ( x X ,Yp),
for X
=
= bA (YX,Yp),
= dxp. From (2.4.8) we obtain
(In), and from (2.4.9), X ~ ) ( W=) 1, b(,)(w) =
for X
=
(n). Of course, in the Schur case (w, X
l / z p wp 1)
X
x ; ( w ) = Y P (w) = x , ,
bx(w) - 1
for all A, p E P,.
PROOF.The proof of this proposition is standard (cf. [5]). Expanding the determinant (2.4.7) by the last column yields
It is also clear that
For X
< p we have LA,
= 0 and
We see that YX = U ' / J ~ ~satisfies both (2.4.3) and (2.4.4), and that bx = ( y X ,y X ) i 2= JXXILXX. Using (2.4.17) again yields
which implies (2.4.16); (2.4.14) and (2.4.15) are straightforward.
88
2. BOUNDARY O F T H E YOUNG GRAPH AND MACDONALD POLYNOMIALS
4.8. Let Xn = xi/^) be the normalized matrix of irreducible characters of the symmetric group 6,. We arrange both rows and columns of Xn in increasing order with respect t o <. The matrix Xn is orthogonal with unit determinant. This fact can be generalized as follows.
PROPOSITION. The matrix Xn(w) =
( d m )
is orthogonal for
all w = (wl,w2 , . . . ) .
For w = 1 these formulas reduce to the ordinary orthogonality relations for irreducible characters. 4.9. Note that hx,, Jx,, u;, and LA, = XuJxuhvxare polynomials in the variables wl, w2, . . . . In order to describe their coefficients explicitly, we need to introduce some new notation. Consider two systems of partitions
from P,, and let PER
PER
Denote by
the minor of Xn corresponding to the rows and columns (2.4.20). For X E P, we write MA= { p : p < A), M A = { p : p A). By the Binet-Cauchy formula,
<
where IR1 = lMxl in both sums. It follows from the definitions of Jx, and LA, that
55. HALL-LITTLEWOOD-MACDONALD
SYMMETRIC POLYNOMIALS
89
where r(X), r ( p ) are the numbers of the partitions A, p, and
In the latter formula (RI = IMP\.
$5. Hall-Littlewood-Macdonald symmetric polynomials 5.1. Recall that A stands for the complex graded algebra of symmetric polynomials in infinitely many variables x = (xl, 22, . . . ), and A,, for the subspace of homogeneous polynomials of degree n . Given a partition X = (XI, X2,. . . ), let PA = p ~ , .p. . ,~where ~ p, = p,(x) = Clcx: is the r t h power sum; the polynomials px, X E P,, form a linear basis in A,. The characteristic map [49,1.71 is the isomorphism of graded algebras ch : F -+ A defined by the formula ch(cP) = pp/zP, p E P. Using this map, one can describe the Schur functions s~ E A as the characteristics of irreducible characters: ~h(~= ' ) sx, X E P. Up to tjhe characteristic map, the definition of GHL-functions in 54 is similar to that of [139].We restate it in terms of the algebra A. 5.2. Define a bilinear form (., .) on A, depending on a sequence w = (wl, wz, . . . ) E Cm, by the formula
where z~ and w~ were defined in 94. Then (px,pL), = SAPfor p i (x) = px(x)/zx WA. In the case w, = 1 we use the simplified notation (., .) = (., .),. Note that the characteristic map is an isometry:
Each GHL-function gives rise t o the symmetric polynomial
called the GHL-polynomial. From (3.4.3) and (3.4.4) we obtain
and these formulas can be used as a definition of the GHL-polynomials. It can be seen from (2.5.3) that the polynomials P A , X E P, form a linear basis in A. The formula
is converse t o (2.5.3); the functions Kx, Kostka coeficients.
=
KxP(w) are called the (generalized)
2. BOUNDARY O F T H E YOUNG GRAPH AND MACDONALD POLYNOMIALS
90
5.3. Consider the endomorphism ww of the algebra A defined by the formula
ww(pn) = (-l)n+lwnpn.
In other words,
where -
&P -
(-l)E(k-l)mk
is the sign of a partition p = (Im1,2 m Z ,. . ). For w = (1,1,. . . ) the map ww coincides with the standard involution w of the algebra A defined by the formula w(sx) = s x ~ X, E P; recall that ( ~ ( f ) ,=~ (f,w(g)). ) Clearly, w;' = wllW.
PROOF.For f
COROLLARy.
= pp, g = p,
we have
( ~ l / W (1 f7 ~ l / W ( g )= ) (f7 g)l/w.
5.4. Using the map wl/,, we will introduce two more families of symmetric polynomials indexed by diagrams X E B:
(2.5.7)
s x =W ~ / ~ ( S X ) ,
(2.5.8)
qX = W I / W ( ~ X ) ,
where ex are the elementary symmetric functions. The following lemma describes these functions more explicitly.
LEMMA.For X = (XI, X2, . . . ) we have qx
= qxl qx,
. . . , where
PROOF.This lemma follows immediately from the formulas
and Lemma 3.3. It also follows from Lemma 3.3 that the functions qx (respectively, Sx) form a dual basis to mx (respectively, sx): (qx, mp)w
= 6xp7
(Sx,sp)w = Sxp.
We emphasize that the definitions of qx and Sx do not depend on the choice of the ordering <. There is another equivalent way to define the polynomials qx, Sx Note that the power sums p l , p2, . . . are algebraically independent and generate the algebra A. Consider the homomorphism cp = cp,,, : A + C defined by the formula
$ 5 . HALL-LITTLEWOOD-MACDONALD
S Y M M E T R I C POLYNOMIALS
91
and write f ( x ) = cp,,,(f) for f E A. Denoting by hx(x) the complete homogeneous symmetric functions, we obtain from (2.5.9) that 4x (x; w) = hA(x) and
Sx (x; w) = 8x (x).
5.5. Define the adjoint GHL-polynomials Q x , X E P, by the formula
then Qx(x;w)
= bx(w) Px(x; w)
for bx = (PA, PA);',
and
An alternative definition in terms of orthogonalization is as follows:
The coefficients in (2.5.13) and (2.5.5) coincide:
5.6. Let us write simply b, for bA with X
The generating function W(z) = ten as
b,zn of the sequence bo, bl, . . . can be writ-
W(z) = exp
(2.5.17)
= (n);then
C zn/nw,. n>l
We will compute the Cauchy kernel
for x = ( x I , x ~. ., .), y = (yl,y2,. . . ) in terms of W(z).
LEMMA.n
( ~Y; ,W) = n,,,w ( x i y j ) .
PROOF.Since
we have
- Y I 2 (fi) Y u l (m) Y g
3-
. s ~ o ~vurura1 ~ o j ayl puv
%)a
(2
a?ON '
(x)Y z u (m) Y
g
3 = (mix)b,
'dOOUd
.vliuvu37
C'z
:l u a p -~!nba axv suo!?!puo3 3u!~o11ojay? ?vy? ysqqslsa ue3 auo ' [ ~ ' P - I' 6 ~ 1u! se Bu!n8.1~ .v vqga31.t. papel3 ay? jo { " ( L ) ' { Y n ) sasvq Jvauq snoaua8ouroy OM?xap!suoD S T V I N O N A T O ~~ T V N O aav ~ ~ H V H~~ X E I~ N ~ O ZH.L A
do
A x v a a n o a 'Z
z6
56. DUALITY
93
56. Duality 6.1. In the previous sections we introduced the GHL-polynomials Px(x;w), Qx(x;w) using the chosen total ordering in the set P,. Consider now the conjugate ordering I' defined in Section 4.3 and denote by P i ( x ; w), Q i (x; w) , bi (w), etc., the corresponding objects. We will use the endomorphism w, from Section 5.3 and denote the sequence ( l l w l , 1 1 ~ 2 ,. . ) by l l w . The main result of this section is
<
THEOREM.The endomorphism w, acts on GHL-functions as follows:
We defer the proof to Section 6.5. In the Macdonald case (M) (see 53) this theorem is equivalent t o Theorem 5.2 from [139]. Indeed, it was shown in [I391 that in this case the superorthogonality property (iii) from 53 holds; therefore in this case P i ( x ; w) = Px(x;w) and Q',(x; W)= Qx(x;w). 6.2. The proof of the duality theorem in [I391 uses some special properties of GHL-functions which hold only in the case (M). Our approach is more general; it is based on the following well-known fact (cf. [57],p. 415): in a unitary matrix with unit determinant each minor is equal to its cofactor.- We will use the notation of Section 4.9. - Let M , R be the complements of the complementary minor, and by the sets (2.4.20) in P,; we will denote by
(g)
{ T} = (-l)r(M)ir(R) (z),the cofactor corresponding t o the minor
(z)
(here
r ( M ) = C r(A) is the sum of the numbers of rows from the set M ; r ( R ) is defined in a similar way). Then
6.3. LEMMA.For all X E
P,,
PROOF. Setting M = { p : p 5' A) and MA, = { p : p (2.4.22) implies
< A'},
we see that
By the definition of the conjugate order we can write M I = {p': p
SIX)
(E)
= { p :p >
A/)
=aX,;
then the minor (?l;i') differs from in two respects: the rows are arranged in the reverse order, and the entries of the columns corresponding t o odd partitions p E P, have opposite signs. Hence
94
2. BOUNDARY O F T H E Y O U N G G R A P H AND MACDONALD POLYNOMIALS
where E ( R )= (2.6.3) that
npER ~ ( pand ) t(R)
Since w R w R =
nw p
=
C p E R ( N- r ( p ) ) , N
=
IP,(. It follows from
= f,(w), the lemma follows.
6 . 4 . L E M M A .For all A, p E P,,
PROOF.Consider the partition P, = M U { A ) U L of the set P,, where M = {p' : p <' A) = { p : p > A') and L = { p : p < A'). Let R range over all subsets in P, with IRI = IMI, and denote by S the complement of the set R U { p ) in P,. According to (2.4.23),
and
Applying (2.6.3) twice yields
Indeed, in order to recover the standard order of the sequence R, p, the column p must be interchanged with ( N - r ( p ) ) other columns. Since f(,)(w)/wS = w , w R , the lemma follows. 6.5. We are now ready to prove the theorem. Let us use (2.4.13) and the formula
( x l ) ; ( w )= w P ( u 1 ) ; ( w ) I f ; ( w ) similar to (2.4.14). Dividing (2.6.8) by (2.6.4) yields
Using the definition of Q i ( x ;w ) similar to (2.5.11) and (2.5.2), we obtain
It remains to substitute w for l l w , and (2.6.1) follows. In order to prove (2.6.2), apply the endomorphism w l / , to (2.6.1),which gives w l l w Q i ( x ; l l w ) = P i ( x ;w ) . This is equivalent to (2.6.2) if we substitute A for A', w for l l w , and Q', P for Q , P ' , respectively. The theorem follows.
56. DUALITY
95
6.6. Let us derive several corollaries from the theorem. As before, we mark functions constructed from the conjugate ordering 5' with primes. For example, K i , ( w ) are the corresponding Kostka coefficients. As follows from (2.5.3) and (2.5.5), the matrix (VX,)is inverse t o the Kostka matrix (Kx,):
PROOF. Applying the endomorphism wl/,to both sides of the formula
PA(( z ;1/11,) =
C vxrp/( 1 1 ~ ) ( 2 ) ~~1
XI',
and using the duality theorem (2.6.2), we see that
(2.6.12)
& i ( x ;w ) =
c
V X / ~ / ( s~p /( xW;w)) .
AS', Since
by (2.5.13), the proposition follows.
LEMMA. b i ( w ) . bxr ( 1 1 ~=) 1. PROOF. Using (2.5.12) and Corollary 5.3 with f ( x ) = g ( x ) = can check that
( x ; l l w ) , one
6.7. Denote by f h ( w ) the structural constants of multiplication in the basis { P A ) of the algebra A:
Let g;V(w) be the analogous coefficients in the basis { Q x ) :
Recall that Q x ( x ;w ) = bx ( w ) P x ( x ;w ) ; comparing (2.6.14) and (2.6.15) yields
2. BOUNDARY OF T H E YOUNG GRAPH AND MACDONALD POLYNOMIALS
96
PROOF. It suffices to apply the endomorphism w, to the formula
and use the duality theorem.
f$(w) COROLLARY.
-
# 0 implies p U v _< X I p + v.
The proof is the same as in Proposition (6.4) in 11391. For w , 1 the functions fk,(w) = g;,(w) = c i , coincide with the ordinary Littlewood-Richardson coefficients c i , 1491, 1.9. It was proved in 11391, (6.6) (in the case (M)) that
f$(w)
#0
implies
X
>p
and
> v.
X
This is not the case for an arbitrary w . 6.8. Consider the special case of the multiplication formulas (2.6.14), (2.6.15) for v = ( 1 ) :
(2.6.18)
p ( i ) ( x ;w ) . PA(x;w ) =
x x
~ A A ( W PA(^; )
w),
A
(2.6.19)
Q ( i ) ( x ;w ) . Q A ( X ; w ) =
M A A ( W ) Q A ( ~w; ) .
A
We call mxi\(w) and M A A ( w )the branching coeficients of GHL-polynomials. By the results of the previous section,
(2.6.20) (2.6.21) (2.6.22)
M A A ( W )= m x ~ ( w. )w l b (~w ) / b x ( w ) M;A(w) = ~ A ~ A ~ ( ~ / w ) , m x A ( w )# 0 implies X U ( 1 ) I AI X
+ (1).
Let us write X 7A if X c A and IAl = 1x1 + 1. We say that GHL-polynomials P x ( x ;w ) for a given w have a Young-type branching if m x A ( w ) = 0 unless X /" A. It follows from 11391, (6.6), that in all cases discussed in $3 the GHL-polynomials have a Young-type branching.
6.9. Let us describe more explicitly the branching coefficients m x A ( w ) for the generalized Jack case (J,). Let X be a Young diagram and s = (2, j ) E A. Following 11391, $7, denote by a ( s ) = X j - j the a m length, by l ( s ) = - i the leg length, and by h ( s ) = a ( s ) l ( s ) 1 the hook length of the square s in the diagram A. Let
+
Xi
+
where a is the parameter of the case (J,): wk,
= a , and
w , = 1 if n # 0 mod m.
PROPOSITION.Denote by mxA(a)the branching coeficients of GHL-polynomials i n the case (J,). Then
where bx ( a )= bx,s ( a ) and the product is over all s E X with h ( s ) = 0 mod m.
57. GHL-FUNCTIONS AND GENERALIZED LEGENDRE POLYNOhlIALS
97
PROOF.This follows from formulas (6.5) and (7.5) of [139] by passing to the limit as described in Section 3.3: q = r cue, t = r E , where r is an rnth root of unity. and E -+ 0.
+
Note that for A
+
< 4 it follows from (2.6.22) that mxA(w)# 0 implies
X /" A
for all w. The simplest example of a non-Young branching is demonstrated by the coefficient rnAA(w)for X = (2'), A = (3 12),which does not vanish for some values of w. For instance, if w = (u, 1,1,.. . ). then
$7. GHL-functions and generalized Legendre polynomials 7.1. Each symmetric polynomial P in n variables x l , . . . , x, can be written uniquely as a polynomial P ( x l . . . . , x,) = R ( e l . . . . , e n ) in elementary symmetric functions e l , . .-. , en. If P is homogeneous, it is uniquely determined by the restriction P = P ( e l , . . . , en) of the polynomial R to the hyperplane en = 1. In particular, for n = 2 we associate with each homogeneous symmetric polynomial P ( x l , 2 2 ) in two variables X I , 2 2 a polynomial ~ ( y in) one variable by setting y = el = x1 2 2 and en = 21x2 = 1. It can be easily checked that for the power sums pn(xl, 2 2 ) = xy x; and complete homogeneous symmetric functions
+
+
we obtain Isn(y) = 2 . T n ( ~ / 2 ) i n (Y) = 2 . Un ( ~ / 2 ) , where Tn(cosP) = cos n p . U,(cos cp) = sin(n 1 ) ~sin / cp are the Chebyshev polynomials of the first and second kind, respectively. Since
+
it is also clear that
Consider now the polynomials q, = q,(x: w) defined in (2.5.9).
PROPOSITION. The polynomial associated with q, ( x l , 22; w) equals
PROOF.Substituting x = ( x l , 22.0, . . . ) in Lemma 3.7, we see that nonzero summands correspond to partitions X = (n-k, k) with two parts. Since B(n-k,k)(w) = bnPk(w)bk(w). the proposition follows from (2.7.1).
2. BOUNDARY O F T H E YOUNG GRAPH AND MACDONALD POLYNOMIALS
98
7.2. Recall that Fejkr [go] introduced the generalized Legendre polynomials P n ( y ) associated with a generating function
as follows:
W ( r e w ).
re-") =
C Rn (cos
(o)
rn.
n>O
More explicitly, n
Rn(cos (o) =
(2.7.3)
bnPk bk cos(n - 2k)(o. k=O
The ordinary Legendre polynomials correspond to W ( z ) = ( 1 - z ) - l l 2
PROPOSITION. Let R n ( y ) = R n ( y ;w ) be the generalized Legendre polynomials determined by the generating function W ( z ) = exp C z n / n w,. Then & ( y ; w ) = Rn ( ~ 1 2 ) . PROOF.
Substituting yl
w (re") . w (re-")
= eip, =
y2
= epiv,
z
=r
into (2.5.23) yields
C qn ( y l ,y2; w ) r n C ?,, ( 2 cos =
(o; w
)rn,
n>O
n>O
and the proposition follows from (2.7.2)
COROLLARY. P ( , ) ( ~w; ) = R n ( y / 2 ; w ) / b n ( w ) , where bn(w) are the coeficients of the series W ( z ) = bn(w) zn. 7.3. Let us consider the special cases enumerated in $3. (S) w n = 1 , n = l , 2 ,.... In this case W ( z ) = l / ( l - z ) , bn = 1, and R n ( y ) = 2Un(y/2) are the Chebyshev polynomials of the second kind. The recurrence relation for these polynomials is
( H ) W, = 1/(1- tn). Here W ( z ) = ( 1 - t z ) / ( l - z ) and bo = 1, bn = 1 - t for n 2 1. Thus R n ( y ) = 2 ( 1 - t ) . [ U n ( y / 2 )- t U n - 2 ( y / 2 ) ] . The recurrence relation coincides with (2.7.4) for n 2, and is 2 y . R l ( y ) = R a ( y ) ( 1 t ) R o ( y ) for n = 1. ( J ) wn 110. In this case W ( z ) = (1- z ) - ~ bn , = (8+:-1), and R n ( y ) are the ultraspherical (Gegenbauer) polynomials. The recurrence relation is
-
>
+ +
where
For 0
=
2 we obtain the ordinary Legendre polynomials.
57. GHL-FUNCTIONS A N D G E N E R A L I Z E D L E G E N D R E POLYNOMIALS
99
(M) w, = (1- q n ) / ( l - t n ) . One can check that in this case
according to the q-binomial theorem [67], bn
-
(t; q), - ( 1 - t ) ( l - qt) . . . (1 - qn-lt) (q;q)n (l-q)(l-q2)...(l-qn)
'
and R,(y) = r, (y; t lq) are the continuous q-ultraspherical polynomials, see [67].
(G) w, = 1 - qn. This case is a subcase of (M). Here W(z) = - qkz), b, = (q; q);', R,(y) = Hn(ylq) are called the continuous q-Hermite polynomials, see [67].
n(l
w, = 118 if n (J,) In this case
=0
mod m , and w,
=
and
1 otherwise.
and the polynomials R,(y) were called in [67] the sieved q-ultraspherical polynomials of the first kind. The coefficients in the recurrence relations (2.7.5) equal n/(O + n - 1) (20+n - 1)/(8+n) 1
(2.7.7)
if n
= km,
if n = km - 1, otherwise.
7.4. It is clear from (2.7.3) that R,(-y) = (-l)"R,(y) for Fejkr polynomials R,(y). If these polynomials are orthogonal with respect to a certain measure on R, then they satisfy a t hree-term recurrence relation of the form (2.7.5). The converse statement is also true, and is known as Favard's theorem: the recurrence relations (2.7.5) with positive coefficients c 0 and initial conditions Ro(y) 1, R l ( y ) = y imply that the sequence R,(y) is orthogonal with respect to a certain symmetric measure on R; see, e.g., [70]. In all cases of Section 7.3 the generalized Legendre polynomials are orthogonal. A remarkable result of Feldheim [60] and Lanzewizky [48] shows that the converse is also true: the generalized Legendre polynomials satisfy the orthogonality condition only in the case (M) (in fact, this condition is also satisfied in the limiting cases (J,), which was not mentioned in [48, 601).
>
--
7.5. Let us write f, = bn/bnPl, n = 1 , 2 , . . . , assuming that bo = 1; then b, = fl f 2 . . . f n Substituting (2.7.3) into the recurrence relation (2.7.5), one can easily check that it is equivalent to the infinite system of equations
< <
15 k n co,for two sequences of variables f l , f 2 , . . . ; cl, cz, . . . . The general solution of this system, which depends on the first three variables f l , f 2 , and f3, was found in [48, 601.
2 . BOUNDARY O F T H E YOUNG G R A P H AND MACDONALD POLYNOMIALS
100
The sequence { e n ) can be expressed in terms of { f,):
so we may go over to the variables f . After some computatiorls which we do not reproduce here (see [GO])one can derive from (2.7.8) that for all n 2
>
Let Y2 = ( f 3 - f i ) / ( f 3 - f z ) , zn = ( f n - f l ) ( f z - f l ) , and assume that for all n 2. Dividing both sides of (2.7.10) by
>
(fn-1
fn
# fn-1
- f n I ( f 3 - f 2 ) ( f 2 - fiI2
yields
(2.7.11)
zn = Y 2 / ( Y 2 - zn-1).
Since zl = 0 and z2 = 1, one can check that
+
where U ~ ( c ocsp ) = sin(n l)cp/sin cp is the Chebyshev polynomial of the second kind. Formula (2.7.12) is valid provided that y / 2 is not a root of any polynomial Un, i.e., y # 2cos7rk/m for integral k , m: These arguments lead to the following result.
PROPOSITION [ 6 0 ] . Assume that f2 # f l , f3 # f l , and
for integral k and m . Define a sequence z l , 22,. . . by the formula (2.7.11)'). Then the sequence (2.7.13)
fi
fn
+ ( f 2 - fi)zn
together with (2.7.9) gives a solution to (2.7.8). The solution (2.7.13) will be called the principal series. There exists another parametrization which uses the variables
9=2
-
f
-
t
-2
= (f2
-
fl)-l[fz(l
- f2)
+(f2
-
fll21.
In terms of these parameters we obtain
y2
=
(9+ l)'/q,
zn = ( 1 + 4 ) ( 1 - ~
~ - ~-)qn), l(l
and hence
Since y
# 2 cos n k l m , q is not
a root of unity.
7.6. Besides (2.7.14), there is another family of solutions of (2.7.8). ')1f
f3 = fi
we set z,
=1
$7. GHL-FUNCTIONS AND GENERALIZED LEGENDRE POLYNOMIALS
PROPOSITION. For given integral m sequences { f n ) and {c,) defined by fi
+(fm
-
>2
and real f l , f,
fl)/k
i f n = km, otherwise
101
# 0, fm # f l , the
and (2.7.9), respectively, satisfy equations (2.7.8).
PROOF.Denoting O
= f,/
f l , write the sequence f , in a more explicit form:
It is clear from (2.7.9) that the coefficients cn are given by (2.7.6), and the proposition follows by direct computation. We will call (2.7.15) the complementary series of solutions of (2.7.8). This fanlily was overlooked in [48, 601; in [65] it appears in another form. The complementary series can be obtained as the limit of solutions of the principal series (2.7.13). For this purpose, denote by r a primitive mth root of unity, and set
Substituting these values into (2.7.14), we obtain (2.7.16) in the limit as S -+ 0. For m = 1 the solution (2.7.16) corresponds to the ordinary Jack case (J). It can be formally included into the principal series (2.7.14) by setting y2 = 4.
7.7. Let us show that the principal and complementary series exhaust all solutions of (2.7.8). To begin with, note that equations (2.7.8) and (2.7.9) are homogeneous; hence the set of solutions is invariant under the transformations
a # 0. Since b, = fl f2 . . . f,, we obtain b p ) = anbn, w ( " ) ( z= ) W ( a z ) , and w t ) = w n / a n for n = 1 , 2 , . . . . It is clear from (2.7.9) that the polynomials R, ( y )/b, are invariant under (2.7.18).
THEOREM. Propositions 7.5 and 7.6 exhaust all solutions of (2.7.8), (2.7.9) up to a nonzero common factor.
PROOF. It follows easily from (2.7.8) and (2.7.9) that
for all 1 < k
< n. Assume that for some k the factor
does not vanish. Then fn+l and c, are uniquely determined by the values f l , f 2 , . . . , f,. We will find such factors for all solutions of the complementary series. Assume that f l = f2 = . . . = fmPl = 1 and O = f , # 1. For k = m a n d s 2 we have: 1. if n = ms, then
>
M = 1 - f m / f,,
=
(S -
1) ( 1 - O ) / ( O
+ s - 1 ) # 0;
102
2. BOUNDARY O F T H E Y O U N G G R A P H A N D MACDONALD POLYNOMIALS
2. if n
= ms
-
1, then
h i ' = 1 - fin . fms-, 3. if n = rns - t , 1 < t
=
(1 - 8) (8
+s
-
l)/(s
-
1) # 0;
< m, then M=1-fm=l-QfO.
It remains t o consider the case m = n, rn 2 3, where we set k = 2. Then 4. M = 1 l / fn = (8 1)/8 # 0. We conclude that the initial values f l = . . . = fin-l # f,, m 2 3, uniquely determine the solution { f,}, and that this sollitiori belongs t o the corriplerrleritary series. Consider now the cases f 2 # f l or f3 # f l . If f 3 = f 2 # f i , then for k = 2 and r~ 3 the factor (2.7.20) does not vanish: M = 1 - f 2 / f l # 0; we obtairl the Jack case ( m = 1). The cases f l = f3 # f 2 arid f l = f 2 # f 3 reduce to those considered in (1) and (4), and deterrrli~iesolutions from the corriplementary series with rrL = 2 and m = 3, respectively. 111 the general case f l # f 2 , f l # f 3 , f 2 # f3 the solutio~i{ fn} is uniquely deterrriined by f l , f 2 , f3 (the principal series), see Section 7.5. The proof is complete. -
-
>
7.8. Let us returri t o symmetric GHL-polyriorriials. It follows from (2.6.22) that
for appropriate coefficients cn = c,(w) = WL(,),(,,~)(W).Assurriirig that x = ( x l , x2,0, . . . ) and using the superorthogonality conditiori (ii) frorri Section 3.4, P(n-l)(x; w), and Corollary 7.2 implies tlie folwe obtain that P(n,l)(:c;w) = lowing:
PROPOSITION. The superorthogonality condition (ii) of Section 3.4 in~pliesthat th,e generalized Legendre polynornials'b Rn(y) associated with symmetric GHL-polynornials sati.sfy the recurrence relations (2.7.5).
>
Assuming that c, 0 for all n = 1 , 2 , . . . and using the rerrlarks from Section 7.4, we can assert that tlie polynomials Rn(y) from the proposition are orthogonal as polynomials in one variable. From tlie last proposition and Theorem 7.7 we obtain our second main result.
THEOREM. Assume that GHL-polynornia1.s Px(x;w) satisfy condition (ii) frorr~ Section 3.4 with n = 2. Then the sequence w(') = {cn w,}, for an appropriate constant c, 2.7 of the form (M) or (J,) (.see Section 7.3). 58. Determinantal formulas for GHL-polynomials
8.1. In the rerrlairider of this chapter we consider GHL-polynorriials only in the Macdonald case (M) and use the notation of [139]:
for wn = (1 q n ) / ( l - t n ) , r~ = 1 , 2 , . . . . We would like to obtain deterrrlinantal formulas for PA,Qx that in the Sclmr case (S) wo~ildreduce to the well-known second Weyl formula: -
58. DETERMINANTAL FORMULAS FOR GHL-POLYNOMIALS
103
From another viewpoint, these formulas generalize the classical representations of orthogonal polynomials by determinants of three-diagonal Jacobi matrices:
+
where c, are the coefficients of the recurrence relations y . R,(y) = Rn+l(y) C , R , - ~ ( ~ )determining the polynomials Rn(y). By continuity, our formulas are also valid in the generalized Jack case (J,). 8.2. Let us introduce the rational functions 1 - qAi-i+j tm-j X cij(9, t ) = 1 - q ~ tm-i i '
where X = (A1, . . . , Am) is a Young diagram with m nonzero rows. We will study the determinants (2.8.4)
m
YA(X; 9, t) = det(c,?j(q,t) e x % - i + j ( ~ ) ) i , ~ = l
and
If q = t, then c $ ( ~t) , = 1, and the determinants cpx = expressions (2.8.1) for Schur functions:
turn into the ordinary
since q,(x; q, q) = hn(x). Following [139], denote by w , , ~ the automorphism of the algebra A defined by
then wt,, is the inverse automorphism. By the duality theorem ([139], (5.2)), wt,, en = ~
t , P, ( l n ) (x; t, 9) = Q(n) (x; 9, t) = qn(x; Q,
t)
and wqVtqn(x;q, t) = en(x). Applying the automorphisms w,,t, wt,, t o the determinants cpx, $A yields
8.3. There are natural generalizations of (2.8.6) and (2.8.7) for hook diagrams.
THEOREM.Let X = (n - k, l k ) be a hook diagram with arm length n and leg length Ic. Then (2.8.10) (2.8.11)
Qx(x; 9, t) = $x(x; 9, t),
PA! (x; q, t) = cpx (x; q, t).
-
k
-
1
2. BOUNDARY O F THE YOUNG GRAPH AND MACDONALD POLYNOMIALS
104
PROOF. Macdonald [139] obtained explicit formulas for the functions b, (q, t) in particular, for p = (1")
= b,(w);
Expanding the determinant (2.8.5) by the first row yields
, t )Q ( l j ) ( ~ ; q , tfor ) all j By induction on n, assume that $ ( l j ) ( ~ ; ~ = follows from Theorem (6.5) in [139] that
< n. It
Hence the j t h term in (2.8.13) equals
All summands in (2.8.13) except Q(n-k,lk)(x; q, t) cancel out, and (2.8.11) follows. Applying the automorphism w,,~and using the duality theorem (5.2) of [139] and Proposition 6.2, we obtain (2.8.10).
The theorem has several interesting specializations; for example, for the HallLittlewood functions (q = 0), monomial symmetric functions (q = 0, t = l ) ,etc. For q = t, formulas (2.8.10), (2.8.15) reduce to (2.8.6), and formulas (2.8.11), (2.8.16) reduce to (2.8.7). For general q, t, and x = (xl ,x2,0, . . . ), (2.8.11) is the determinant of the representation (2.8.2) for Rogers polynomials. 8.4. If a diagram X is not a hook, the formulas become more complicated. For example, for X = (a 1,2, lb-l)
+
where (q - t) (1 - ta) (1 - qb-lt) (1 - qbta+l) (1 - t) (1- qta) (1 - qb-92) (1 - qbta-1) ' Recall the definition of the F'robenius parametrization of a diagram X E P. Denote by d = d(X) the largest i such that Xi 2 i, and let fi = Xi - i, gi = A', - i for i = 1 , 2 , .. . , d, where A' stands for the conjugate partition of A. Then the Frobenius symbol ):; uniquely determines A.
K=
(i:::::
59. BRANCHING O F MACDONALD POLYNOMIALS
105
+
Denote by f&g = (f 1, l g ) the hook diagram with arm length f and leg length g. There is a well-known expression for Schur functions in terms of hook Schur functions sfhg(x):
CONJECTURE. For all X
E P there exists a rational function MA = h'fA(q,t)
such that
for all x = (x1,x2,.. .). For example, for X
and for X
=
(a
=
(a, b)
+ 1, 2, lb-')
59. Branching of Macdonald polynomials 9.1. The classical Hall-Littlewood polynomials have numerous applications in representation theory ([49], [143]). For example, consider GHL-functions X i , Yk in the Hall case (H) as t 4 -1. Then the characters of projective representations of the symmetric group B,, found by Schur, can be written as
where X is a strict partition of n with m parts; &(A)= 0 if ( n - m) is even, and &(A)= 1 otherwise. Here p ranges over all partitions with odd parts which index the conjugacy classes of the second kind in 6,. The second example is the Hecke algebra H,(q) of the symmetric group 6,. It was shown in [19] that the characters of H,(q) can be described in terms of symmetric functions by a formula similar to Frobenius' formula:
where P = (pi,p2,. . .), rp = rplr p z . .. , and r, = r, (x; q) = qm-'
. P(,) (x; 0, l/q).
We denote by X; the values of the irreducible characters xX,X E P,, on the elements of the form Cp
where t l , . . . ,t,-l
= (tl...tpl-l)(tpl+l...tP1+PS-l).
are the standard generators in H,(q).
. .. ,
2. BOUNDARY O F T H E YOUNG GRAPH AND MACDONALD POLYNOMIALS
106
9.2. Another point of view on symmetric functions arises from the description of the characters of factor representations of the infinite-dimensional algebras @[6,]and H,(q). According to Thoma's result [161], such a character x,,~is indexed by a pair of nonnegative sequences
>
>
>
> P2 2 . . . 0, and y = 1 Cai Cpi > 0. > 2, equals
such that a1 aa ... 0, ,Bl value of x,,p on the cycle c(,), m
-
-
The
A similar result was obtained in [19] for the Hecke algebras H,(q):
where P(,)(a,p; 0,114) is defined by substituting pn = p n ( a , P) into (2.5.2). For the so-called Markovian traces trq,t of the algebra H, (q) which were studied in connection with an algebraic construction of Jones' knots invariants [114], we have
where the coefficients are related to Macdonald HL-polynomials as follows:
If b E Hn(q) c H,(q), then the sum in (2.9.2) is over X E P N for any N 2 n. The scalar product (2.5.1) and the corresponding GHL-functions in the case (M) can be related t o representations of quantum groups and their centralizer algebras, see [148].
6,
9.3. The description of the characters of factor representations of the group = l q 6, reduces to the following question:
(*)
Find all homomorphisms cp : A
-+
@ such that cp(sx)
> 0 for all X E P.
A homomorphism cp can be defined by the generating function
Then the answer to (*) is as follows:
where a , p are the same as in (2.9.1). Consider a more general problem:
(**) Find all homomorphisms cp : A where PA= PA(x; q, t).
-+
@ such that cp(Px)
> 0 for all X E P,
59. BRANCHING O F MACDONALD POLYNOMIALS
107
CONJECTURE. T h e generating functions (2.9.3) for homomorphisms that are positive i n the sense of (**) are of the form
where
One can show that all these homomorphisms are indeed positive. In some cases this list is known to be conlplete: 1. in the Schur case (S) this was proved in [161, 131. 2. for projective characters of the group 6, (q = 0, t = -1) the conjecture was proved by Nazarov [51]. 3. in the case q = 0, t = 1 the conjecture follows from Kingman's result [130]; see also [34].
CHAPTER 3
The Plancherel Measure of the Symmetric Group In this chapter we study the properties of the Plancherel measure of the infinite symmetric group. For the sake of completeness, in $1we reproduce the results of A. M. Vershik and the author [12, 181 on the typical limit shape of large Young diagrams. These results were the origin of a considerable part of this work. In $2 we obtain an analogue of the central limit theorem for the characters of 6, which is a refinement of the main theorem of the first section. In 53 the theorem on the limit shape is used t o study the statistics of symmetry types of random tensors. In $54 and 5 we consider a q-analogue of the hook walk algorithm, and in 56 we compute the multivariate Selberg integrals as generalized Poisson integrals for appropriate branching graphs. 51. The typical shape of random Young diagrams
Vershik and the author [18] obtained order-sharp two-sided bounds on the typical and maximum dimensions of irreducible representations of the symmetric group G N as N -+ CO. Both problems were solved simultaneously and related to the theorem on the limit shape of typical Young diagrams which had been proved earlier in [12]. Since these theorems provide the motivation for many results of this book, we will reproduce their statements and sketch the proofs. 1.1. The main results. Let 6~ be the symmetric group of order N and denote by bN E yN the set of all equivalence classes of its complex irreducible representations. Given A E &iN, denote by dimA the dimension of A. The problem of computing the maximum dimension dim(A) was posed long ago (see [141]).Recall that by Burnside's formula
dim2 A = N!; A E ~ N
eN,
hence dimA < n ! for all A E and the natural normalization of dimension is d i m ~ l n ! .It was conjectured that there exist gigantic representations with d i m A / n ! 1/N. This conjecture was based on the numerical data obtained in [141] for N 5 75 (!); later it turned out that it fails for N = 81. However, one could think that m a x d i m h l f i ! 2 P ( N ) - l , where P is a polynomial. The next theorem disproves this conjecture and shows that the quotient m a x ~ dim A I f i ! decreases as e x p ( - c n ) , i.e., faster than any polynomial.
>
3.
110
T H E PLANCHEREL MEASURE OF T H E SYMMETRIC GROUP
THEOREM A. There exist positive constants c0 and cl such that, for all N = 1,2, ...,
The problem of computing the maximum dimension turned out to be very closely related to another problem, that of the typical dimension. Given A E kN, put pN(A) = dim2 AIN?. It follows from Burnside's formula that p~ is a probability measure on kN;it should be called the Plancherel measure. Note that the Plancherel measure is naturally distinguished; p,(A) is the relative dimension of the isotypic component of the representation A E E N in the regular representation of e N . It is with respect to this measure that statistics and asymptotics of characters should be studied. It turns out that the asymptotics of the typical (with respect to the Plancherel measure) dimension coincides in order with the asymptotics of the maximum dimension.
THEOREM B. There exist positive constants cb, ci such that N-oo
A : cb <
2 - ----
JN
dim A log -< c ; } = 1.
m
I n other words, dim2 A
=
1- o ( l ) ,
AEFN
where FN= {A E kN: m e - * m
< dim <
The interaction of Theorems A and B is very remarkable. In [12] (see also [138])a lower bound on the typical dimension, and hence on the maximum one, is obtained; a slightly coarser lower bound on the maximum dimension is due to McKay [141]. The limit shape R of the typical Young diagram is also found in [12] and [138]. It would be natural to conjecture that the diagrams of the representations of maximum dimension also converge, in an appropriate scaling, to the same curve. In [18]it was shown that this is indeed the case. Moreover, this yields an upper bound on the maximum dimension, and hence on the typical one. Unexpectedly, the logarithmic orders of the maximum and typical dimensions coincide (the difference being only in the constant), but it is this unexpected fact that relates both problems. We do not pretend that the constants in Theorem B are exact. We prove only that one can take cb = c0 = 0.2313 and ci = cl = 2.5651. However, Theorem B is likely t o admit the following strengthening: there exists a constant c (the entropy of the Plancherel measure) such that for all E > 0
Moreover, the limit dim A lim log -= c N+oo
JN!
seems to exist for almost all infinite Young tableaux (in the sense of the Plancherel measure on tableaux defined in Section 3.1 of the Introduction). These statements would mean that the Plancherel measure is asymptotically equidistributed and give
51. T H E TYPICAL SHAPE O F RANDOM YOUNG DIAGRAMS
111
an analogue of the Shannon-Macmillan-Breiman theorem, the constant c being the "specific entropy of the representation". This would not be quite the usual Shannon theorem: there is a Markov chain with a fast-growing number of states, and we wish to prove that the states at the moment N are asymptotically equidistributed as N + oo. Numerical experiments confirm this conjecture and the bound c > 1.8. The proof goes essentially along the following lines: we transform the formula for the dimension, then solve a variational problem, and prove that the solution is unique. For this purpose, the following expression for the hook integral was used in (121:
1 f 1'
= -//log21s
-t
. f f ( s ) . f l ( t ) dsdt.
These considerations allowed one to investigate the asymptotics of the shape and dimension of the typical diagram with respect to the Plancherel measure, but gave no information on the maximum dimension. A substantially new idea was to write the quadratic part of the hook integral in the form
11 f 1 '
=
// (f
-
s-t
(t))
' ds dt.
The appearance of the Sobolev norm and Hilbert integral in a combinatorial problem concerning Young diagrams of maximum dimension looks surprising. The proof of Ulam's conjecture was announced in [12] as an application of the theorem on the limit shape. It requires some new considerations compared with Theorem B; namely, we need an additional upper bound on the length of the first row of the random Young diagram. We give a detailed proof in Section 1.4. 1.2. The hook integral. The proof of Theorems A and B is based on the remarkable hook-length formula of Frame, Robinson, and Thrall [92]: dimh
N!
=-
n hi,
'
Here the product is over all squares (i,j) of the diagram A, and hij = Ai+Al-i-j+l (where is the length of the j t h column) is the number of squares in the hook with vertex (i,j ) . This formula is equivalent to the well-known Frobenius formula, but it is much more convenient because of its symmetry and multiplicativity. It implies the following formula for the Plancherel measure:
Xi
Taking logarithms, dividing by
where JA = 1
fl,and applying Stirling's formula yields
+ $ Czjlog d'z and
depends only on N
112
3.
THE PLANCHEREL MEASURE OF THE SYMMETRIC GROUP
Let y = F ( x ) be a compactly supported bounded nonincreasing function dey). Denote by h ~ ( xy), = F ( x ) fined on [O, w ) . Set F-' (Y) = inf{F(x) F-l(y)- x - y the hook at the point (x, y), and let SF = {(x,y) : 0 y < F ( x ) , 0 5 x < w) be the subgraph of F . The hook integral of the function F was defined in [la]as the double integral
<
<
+
Given a Young diagram A with N squares, denote by A the subset of R2 obtained by scaling A by fi.Let us call A the normalized diagram; it has unit area. Let F = FAbe the function with SF = A; then JAis an integral sum for the hook integral OF. Below we will prove that if FA, -, F for a sequence of diagrams A1, A2,. . . , then lim
N-oo
1 log p N (AN) = dF. N
-
EXAMPLE.Let AN be the triangular diagram; the normalized diagrams converge to the subgraph of the function F ( x ) = fi - x, 0 x 5 fi.Then
<
dim AN =
AN
a!exp(-N-- 2 J3Z + o(N)),
and p N (AN) decreases exponentially fast, In a similar manner formula (3.1.4) can also be used for computing the asymptotics of dimension for other curves except one: for the "arcsine law" R (which is the limit of both typical normalized diagrams and those of maximum dimension) the right-hand side of (3.1.4) vanishes: On = 0. It turns out that to study the asymptotics of dimension in this case one should take another scaling, -w m ' and pN(AN) decreases subexponentially. In order to estimate the deviation of the integral sum JAfrom the hook integral, set B(A) = (JA- O A ) f l ; then
where
in particular, G(A) > 0 for all A E YN Let us reduce the hook integral t o the quadratic form. To this end, introduce the coordinates X = i ( x - y), Y = $ ( x + ~ )in , which the boundary of a normalized diagram is the graph of a function which will be denoted by LA. This function is a rectangular diagram, i.e., it is piecewise linear, continuous, and 1) L',(X) = f l ; 2) L A ( X ) 2 1x1 and L A ( X ) = 1x1 for sufficiently large 1x1.
$ 1 . T H E TYPICAL S H A P E OF R.ANDOM YOUNG DIAGR.AMS
Then A = { ( X , Y ): (XI 5 Y 5 L ( X ) ) . Given an arbitrary piecewise smooth function Y
0 ( L )= 1
(3.1.6)
+ 2 ]i<s[log2(s
-
=
L ( X ) ,set
t ) ](1 - L 1 ( s ) )( 1
+ L 1 ( t ) )d d t .
LEMMA2. Oh = @(LA)for all A E YN. Let us find the critical point of the hook integral and its first variation at this point. To this end, we use the following version of the arcsine law:
if condition 2) holds, then f is a co~npactlyslipported Let f ( X ) = L ( X ) - fi(x); fiinction. LEMMA3 . The hook integral can be represented in the form
(3.1.8)
0 ( L )= - / / 1 0 g 2 ~ ~
-
t f 1 ( s ) f ' ( t ) d s d t+ 4
f ( s )arccosh Is1 ds.
It turns out that the quadratic part of the hook integral coincides with the Sobolev norrn
1 f l 2Q
-
//
( f
3-t
dsdt
in the space of piecewise smooth compactly supported functions: LEMMA4.
0(fi + f
)
=
il l f 11; +
4
f ( s )arccosh Is1 ds.
COROLLARY. If L ( s ) 2 s and L # f i , then 0 ( L ) > 0; thus strict minimum of 0 (.) on the set of admissible functions.
R
is the unique
1.3. Proofs of the theorems. A lower bound on the hook integral follows frorn the following result. LEMMA5 . If A E inequality
YN and
f ~ ( s= ) L A ( s ) - R ( s ) , then for every
E
> 0 the
holds for suficiently large N
PROOF.The points si = i / 2 f l , i E Z, divide the line into intervals of length A s = 1 / 2 m on which the function L A is linear. Assurne that the rninirnum value of [ f L ( s ) l 2for .7i < s < s i f l is attained at s5. Then
3.
114
T H E PLANCHEREL MEASURE O F T H E SYMMETRIC GROUP
for all s, t E [si,s ~ + ~i ]E, Z; taking the integral in (3.1.9) only over the union of i E] Z, , we obtain the squares s , t E [si,s ~ + ~
Replacing the integral sum in the right-hand side by the corresponding integral and using Lemma 4 yields
for every
E
> 0.
COROLLARY (an upper bound on the dimension).
or, which is the same,
max dim A
< fle-"ofi.
For co one can obtain the bound 2
I signs - f i ' ( ~ ) ds 1 ~= -(7~ 7T2
-
2) E 0.2313.
Let us now give a lower bound on the dimension of typical diagrams. Let
Diagrams from the set MN will be called essential.
PROOF.The total number of diagrams with N squares is traditionally denoted by p ( N ) ; by the Euler-Hardy-Ramanujan formula,
Hence if A E YN \ M N , then PN(A) p ( N ) . e - % O + 0.
<e
by (3.1.5), and PN (YN \ MN) 5
Thus we have proved both bounds, and Theorems A and B follow. Note that McKay [141] applied similar arguments to obtain a lower bound on the maximum dimension. Instead of Burnside's formula he used the identity CAdim A = t N , where t N const . ( F ) N / 2. efi is the number of involutions in e N . It f0110ws that 1 tN -max dim A 2 - const. e-(%-l)fi.
dm
PN fi
$1. T H E TYPICAL SHAPE OF RANDOM YOUNG DIAGRAMS
115
%
Our bound is slightly more exact: = 1.2825 < 1.5651 = 3 - 1 (see Section fi 3.4). Let us now prove the theorem on the limit shape of essential diagrams. THEOREMC (cf. [12],[138]).If A E M N , then
PROOF. First of all, let us estimate the L2-norm of the difference f A ( s ) = LA(s)- O(s). Choosing an interval [-a, a] that contains the support of f A , we may decompose the integral (3.1.9) into two summands, (3.1.10)
lf~ll;=
1:la (f ( s ) f ( t ) ) 2 S-t
dsdt + 8 L
f2('lds a2 - s2
'
Hence, for an essential diagram A E M N ,
Since I fA(s)J norm:
< 2 for every Young diagram,
we obtain a bound on the uniform
and the theorem follows: (1 f A11 L , 5 C . N-lI6 1.4. Solution of Ulam's problem.. Let us show that for an essential diagram A E YN the length of the first row rl(A) has asymptotics rl(A) 2 a . Theorem 2 1 - C - N-lI6, i.e., it gives only a lower bound. In order to C implies that obtain an upper bound for rl(A), let us find the asymptotics of the mean value of rl (A) with respect to the Plancherel measure.
$$
-
PROOF. Let T be the space of (infinite) Young tableaux, and denote by p the Plancherel measure on T (see Section 3.1 of the Introduction). Then /I is a Markov measure; if a Young diagram A E yk is obtained from a diagram X E y k P 1 by attaching one new square, then the transition probability equals p ~ , h= &. We will use the following typical identity for the Plancherel measure:
Denote by A' E yk the Young diagram obtained from X E yk by attaching one square t o the first row, and let Gk be the indicator function of the set of tableaux t = ( X I , X 2 , . . . ) E T with Xk = The mean value ($k) of gk with respect t o the Plancherel measure equals ($k) = CAEYk-l pk-1(X) p x , ~ , so, ; taking (3.1.11) into account, we obtain
3.
116
i.., (
T H E PLANCHEREL MEASURE O F T H E SYMMETRIC G R O U P
) < A. Since r1 ( A N )
=
N Ekzl $lc(t),we finally obtain
(%)
n/fi
By Lemma 6, < 2. On the other hand, by Theorem C, > 2-E almost surely with respect to p~ as N -+ cc for every E > 0, and we have proved
for every
E
> 0.
Of course the number of columns el (A) of the typical diagram A E YN has the same asymptotics, cl(A) 2m. Theorem D provides a solution of Ulam's well-known problem. Let 5 = {tk)Y be a sequence of i.i.d. random variables with a common continuous distribution. Denote by R1 (E) the length of the longest increasing subsequence in I .
COROLLARY. For every
E
>0
PROOF. The Robinson-Shensted-Knuth algorithm (RSK) (see [152, 157, 133, 1271) defines a map from the space of sequences into the space of tableaux T. It maps a product measure with a continuous factor to the Plancherel measure. Moreover, by a well-known property of the RSK-algorithm, the length of the longest increasing subsequence coincides with the length of the first row of the corresponding Young diagram (Shensted's theorem, see [157]). The corollary follows now from Theorem D. Shensted's theorem admits a generalization: the number of squares in the first k rows of a Young diagram is equal to the maximum number of elements in the union of k monotonically increasing subsequences. This fact leads to a strengthening of Theorem D, obtained in [la],which we omit here. 52. Gaussian limit for the Plancherel measure In this section we show that the Plancherel measures of the symmetric groups + cc to a Gaussian random process in an infinitedimensional linear space. With respect to the law of large numbers for these measures, which was found earlier in [12, 181, this statement is an analogue of the central limit theorem. The idea of such a theorem was discussed in [138].
6, weakly converge as n
A
2.1. Main results. Let 6, be the symmetric group of degree n , and let 6, be the set of (equivalence classes of) its irreducible representations; the character of a representation X E 6, will be denoted by x'. Given a partition p = (1'12'2 . . . ) of n , consider the class of conjugate permutations in 6, having rk cycles of length k for k = 1 , 2 , . . . , and denote by the value of the character XX on this class. A
Xt
$2. GAUSSIAN LIMIT FOR T H E P L A N C H E R E L MEASURE
117
A
Let dim X = X&,) be the dimension of a representation X E 6,. By Burnside's theorem Ex dim2 X = n!, so that the weights Mn(X) = dim2 Xln! form a probability distribution Mn on the set g,, called the Plancherel measure of the group 6,. Define the random variable
as the (normalized) value of characters of representations X on the class of permutations with a single nontrivial cycle of length k. The main result of this section is the following:
THEOREM 1. For every x2,x3,. . . , x, (3.2.2) lim Mn{X E 6 , : pk(X)< x i , 2
n-00
E R there exists the limit
< k < rn) = k=2
In other words, the functionals cpk(X)are asymptotically independent and have Gaussian limit distributions with zero mean and variance k . Let us give alternative statements of Theorem 1 related to other systems of basic functionals. We will use the standard identification of the set 6, with the set y, of Young diagrams with n squares. Introduce the following notation for the unital Chebyshev polynomials of the first and second kinds on the interval [-2,2]:
+
sin(r 1 ) O sine '
t T ( 2 c o s 8 ) = 2 c o s r 0 , u,(2cosO)=
Let d be the length of the main diagonal of a Young diagram X = (A1, X2,. . . ) E Yn, and denote by iii = (Xi - i)/,,h, bi = (Xi - i ) / f i , i = 1 , . . . , d, its normalized Frobenius coordinates. For r = 2 , 3 , . . . , set (3.2.4)
tT(x)=
x
+
(tT(iii) (-I),+'
tT(ii)).
Z
We will denote by (f), = CxEyn f (A) Mn(X) the mean value of a function f with respect to the Plancherel measure Mn. THEOREM 2. The Plancherel mean value of the random variable (3.2.4) vanishes for even r and equals
f o r r = 2m+ 1. As n i a, the centralized variables :,(A) = t,(X) - (t,),, X E yn, are asymptotically independent and normal. The limiting variance t,(X) is equal to lim(;:),
= r.
Let c(i, j) = ( j - i)/,,h be the normalized contents of the square (i, j) of a Young diagram A. For r = 1,2, . . . , set ur(X) =
1 -
C
Jii ( i . j ) E *
ur(,).
j-i
118
3.
T H E P L A N C H E R E L M E A S U R E OF T H E S Y M M E T R I C G R O U P
THEOREM 3. The Plancherel mean value of the random variable (3.2.6) vanishes for odd r and equals
for r = 2m. A s n + cm, the centralized variables &(A) = u,(A) - ( ~ r ) n A, E Yn, are asymptotically independent and normal. The limiting variance ur(A) is equal to lim(gT)n = l / ( r 1).
+
Let us now regard Young diagrams as piecewise linear continuous functions w with derivative f1 (this viewpoint goes back to [12]). For instance, the one-square diagram w E yl is identified with the function w(x) = max(lxl,2 - 1x1). It follows from the law of large numbers established in [12] that the Plancherel mean value
of a random diagram with n squares (scaled by ,/6 along both axes) uniformly converges to the function
Embed the set yn of Young diagrams with n squares into the space of functions by associating with a diagram w E y, its-deviation Aw(x)from the curve (3.2.8); more precisely,
Identify the Plancherel measure Mm with its image under this embedding.
THEOREM 4. AS n + cm, the measures M, weakly converge to a Gaussian random process o n the interval [-2,2]. The limiting process has zero mean and covariance function
Substituting x = 2 cos cp, y = 2 cos $ and using the definition (3.2.3) of Chebyshev polynomials, we can write the function (3.2.10) more explicitly: B ( 2 cos cp, 2 cos $) =
1
2.2. Proof of Theorem 1. Let y = Un>oyn be the set of all Young diagrams (or, equivalently, of all partitions of positive integers). With each partition p E y we associate the family of central elements c,;, E Q[G,], n 0, defined by
>
-
Here p = (Ir1,2rZ,.. . ) is a partition of the number r = Ipl = k r k , C(p,ln-r) is the uniform probability distribution on the conjugacy class C(p,ln-r) of G,, and (n), = n ( n - 1) . . . ( n - r 1). For n < r we set c,;, = 0. It is remarkable that in the basis (3.2.12) the convolution is stable in n:
+
$2. GAUSSIAN L I M I T F O R T H E P L A N C H E R E L M E A S U R E
119
PROPOSITION 1. For each pair of partitions a, T there exists a (unique) finite family of integral nonnegative coeficients f$,, such that
for all n
> 0.
Without reproducing the proof, we restrict ourselves to a clarifying example: C(2);n
* c(2);n= c(22);n+ 4~(3);n+ 2 ~ ( 1 2 ) ; n
In the basis of conjugacy classes the coefficients of expansion depend on n:
Define the class ring C as the free abelian group with generators {cp), p E and the multiplication determined by the structural constants f$,,. The elements cp E C determine functionals on G = Un,o Gn by the formula cp(A) = X X ( ~ p ; ndim ) / A. It is easy to see that
y,
A
(3.2.14)
(c,
Let deg(cp) = ri
* cr ) (A) = C,
+ /pi. If f:,,
(A) C, (A),
# 0 in
A
AE
6.
(3.2.13), then
Thus we have defined a filtration on the class ring; we will need a detailed description of the corresponding graded ring gr C. By definition, this ring is additively generated by the same basis {c,}, and the multiplication is defined by the highest degree terms: deg p=deg a+deg r
In the formulas below, products and powers of elements cp are taken in the ring grC, i.e., up to terms of lower degree. The elements ck = c(k) corresponding to partitions p = (k) with a single part will be called the fundamental classes.
PROPOSITION 2. The elements cl, c2,. . . are algebraically independent and generate the ring gr C Z[cl, ca, . . . 1. If partitions a, T have no equal parts of length greater than one, then c, c, = c,",, where a U T is the union of the parts of a and T . If all parts of a partition are equal to k, then
--
(3.2.17)
C(kr)
where H r ( s ) =
=
SL H r ( ~ k / ~ k ) ,
ez2l2( e - ~ ' / ~ ) ( ' )are the Hermite polynomials and sk
The proof is based on the identity
where
r~
is assumed to have no parts of length k. By induction we obtain
=
@.
120
3.
T H E PLANCHEREL MEASURE OF T H E SYMMETRIC GROUP
in the ring gr C. The irlversiorl of this formula gives (3.2.17). Let us show that the Plancherel mean values (f), for f E C are polynomials in n. LEMMA.
(Cpf p ~ p ) n= C
k
f(lk)
. (n)k.
The proof follows from the orthogonality relations for characters of the symmetric group:
It follows from this lemma and (3.2.19) that
and (ci), = 0 for odd r ; moreover,
) / ~ p E Y ; i r ~particular, the variables pk = p(k) are Let pp(X) = n ~ ~ " g ( " pcp(X), equivalent to those defined by (3.2.1). Denote by ( p y p y . . . jCx3= l i m ( p 7 p T . . . ), the limiting momerlts of the Plancherel measure. Orle can see from (3.2.20), (3.2.21) that
Icm (2m - l)!! if r
= 2m,
ifr=2m-1, and
Thus the momerlts (3.2.22) coincide with the moments of the normal distribution with variance k , and (3.2.23) implies that the random variables p2,p3,. . . are independent with respect t o the limiting measure. Theorem 1 is proved. 2.3. Proofs of Theorems 2-4. Consider the modified F'robenius coordinates defined by ai = Xi - i 112, bi = X: - i 112. According to Lemma 1 from [13], the supersymmetric Newton sums
+
+
in the coordinates ai, bi of a Young diagram X belong t o Q[cl, c2, . . . ] and generate it as a ring. It is easy to check that the Newton sums
in the contents of squares of X generate the same ring.
53. DISTRIBUTION O F SYMMETRY TYPES O F HIGH RANK TENSORS
121
PROPOSITION 1. The normalized fundamental classes (3.2.1) can be expressed i n t e r n s of the sums (3.2.24) as follows:
where 62m-1
= 0,
+
62m = ( - l ) m / m ( m I ) , and o(1) refers t o the L2-norm.
Let us compare (3.2.26) with the explicit formula for the Chebyshev polynomials of the first kind:
Theorem 2 follows now from Theorem I. Theorem 3 is equivalent t o Theorem 2, since p,+l = ( r 1)T, up t o terms of lower degree deg. In order t o prove Theorem 4 , observe that the expression
+
is an integral sum for the double integral
'S
u,(x) d x d y = 2 where that
u,(x) ( G ( x ) - 1x1) d x ,
Dw= { ( x ,y ) : 1x1 5 y 5 G ( x ) ) and G ( X )= w ( x f i ) / f i . It is easy t o check
Therefore
where A,(x) is defined by (3.2.9). Theorem 4 now follows from Theorem 3. 53. Distribution of symmetry types of high rank tensors = The symmetric group G N acts in the space C m of rank N tensors over m-dimensional space by permutations of factors. Denote by yN the set of all Young diagrams with N squares, and by yN,,, the subset of diagrams with a t most m rows. The space decomposes into the direct sum of primary components under the act ion of G :
V N ,=~
@
VN,m(A).
A ~ Y N , ~
Tensors from V N , ~ ( Aare ) said to have symmetry type A. Denote by ~ N , ~ ( =A dim ) VN,,(A)/ dim VN,, the relative dimension of the primary component VN,,( A ) . The numbers pN,m ( A ) determine a probability measure on the distribution of symmetry types. The purpose of this section is to describe the asymptotic behaviour of p ~ , ,as N -+ cc in two cases: when m .v N / y , y = const, and when m = const.
$3. DISTRIBUTION OF SYMMETRY TYPES OF HIGH RANK TENSORS
123
PROOF. Let A' be the Young diagram with ( N - XI) squares obtained by removing the first row from A. Clearly, dim X 5 dim A', and hence
(E)
pN(h)= Let p
= X i IN,
9
5 (N)
N!
X i Xi!(N - Xi)!
q = 1 - Xi/N; the inequality
.-
< 1 implies
(E)pXlqN-X1 -
i.e., -q log q 5 p. Using the bound XI! 2 (Xl/e)'l and taking logarithms, we obtain
<
l o g p ~ ( X ) log
since Np = XI and Alp
=
(c)
- log XI!
( ~ ~ I f l )Lemma ~ . 2 follows.
Lemmas 1 and 2 imply COROLLARY 1. pN,m(X) where y = lim~,,
< exp (-a/2(loga
-
1) - y / 2 j f i ) ,
N m .
3.3. The main theorem. Following [12], scale a Young diagram X E
YN by
fialong both axes (so that the rescaled diagram will be of unit area) and consider
the coordinates X = i ( x - y), Y = i ( x + y). Then the border of the diagram is the graph of a piecewise linear continuous function Y = L x ( X ) . THEOREM1. Let l i m ~ - ~= y. Then, for every
E
lim ~ N , ~ E { YN X : sup ILx(X) - R(X)I X
N-a,
> 0, < E)
=
1.
PROOF. Consider the sets of diagrams
DN(c) = {A E
YN : ~ N ( X2) exp - c f i ) ,
where c E R. It follows from the well-known Hardy-Ramanujan bound p(n) = o(exp 2 x m ) on the number p(n) p~,,-measure:
=
lYN 1 of all partitions that D N , is~ of asymptotically full
pN,m ( Y N ,\~V N , ~ ) 0 as N Let us show that VN,mc VN(C)for sufficiently large c. +
+
0.
$3. DISTRIBUTION OF SYMMETRY TYPES OF HIGH R A N K TENSORS
125
THEOREM 2. AS N 4 cc (with m fixed), the measures P N , weakly ~ converge to the absolutely continuous measure Pm i n the cone Cm with density
where
is a normalization constant.
PROOF.Let be the joint distribution in Hm of the reduced row lengths 21,. . . , xm with respect to the multinomial distribution
By the Moivre-Laplace theorem ([25], $13) the measure the Gaussian distribution in Hm with density
weakly converges to
The formulas of Section 1 yield
-
1 1 . .. ( m - I)!
ncXi
-
xj)2~(m-l)m/2
(N/m)(m-l)m/2
Theorem 2 follows. The measure Pm arises in various problems (see [149, 1421). It is easy to obtain the most probable values of reduced row lengths: LEMMA4. The function cpm(x) attains its maximum value i n the cone Cm at -
the vector z = J $ ( z l , . . . , z m ) , where polynomial Hm ( 2 ) .
21,.
. . , z m are the zeros of the Hermite
The proof reproduces the proof of Stieltjes' theorem, see [56], Section 6.7. COROLLARY 2. Assume that the dimension m of the ground space is fixed and let N -+ co. The space VN,,(X) of tensors of symmetry type X has the maximum
where z l , . . . , zm are the zeros of the Hermite polynomial
For the finite-row analogue of the Plancherel measure a close result was obtained in [68].
126
3.
T H E P L A N C H E R E L MEASURE O F T H E SYMMETRIC G R O U P
$4. A q-analogue of the hook walk algorithm Greene, Nijenhuis, and Wilf [102], [103] suggested a remarkable probabilistic algorithm, the hook walk. This algorithnl was used for two purposes: for proving the hook-length formula and for generating randorn Young tableaux distributed according to the Plancherel measure. In this section we present new applications of the original Greene Nijenhuis Wilf algorithm and describe a y-analogue of this algorithm arid two of its applications. The first application is related to an interesting family of distributions on (infinite) Young tableaux which should be regarded as a natural y-deformation of the Plancherel measure. These distributions are in a one-to-one correspondence with Markovian traces on the Hecke algebra which arise in one construction of Jones' invariant for topological knots arid links. The new algorithm allows one to generate raridonl Young tableaux with these distributions efficiently. The second application is related t o a new interpretation of the kriown qanalogue of the hook-length formula. It turns out that one can assign rnultiplicities (rational functions in q) to the edges of the Young graph so that the recursively defined "y-dimension" of the Young diagram will be given by the q-hook-length formula. For y = 1 all rnultiplicities are equal to one, arid the proof yields the classical formula. Let us reproduce some kriown facts concerriirig the hook-length formula and the Plancherel measure, using the riotatiori and ternlinology of [49]. 4.1. The hook-length formula and transition probabilities. Given two Yourig diagranls A, A, we write X /" A if X c A arid IAI = ( X I 1, i.e., A has exactly orie square rnore than A. The relation X /" A provides the set y of all Young diagrams with the structure of a directed graph (the edges of this graph being pairs (A, A) with X /' A). Directed paths of this Young graph exiting from the initial vertex A = 8 are called Young tableaux. The dirnensior~of a Yourig diagram X is the number fx defined recursively as follows: fo = 0 for the enlpty diagrarn X = 0, arid
+
It is clear from the definition that fA is the number of Young tableaux of shape A. It is also knowri that fx coincides with the dimerisiori of the corresporiding representation of the synlrnetric group. According to the classical hook-length fornlula [92], fx = n!/ h i j , where hij is the hook length of the diagram X at the square ( i ,j ) . In this context, the Plancherel measure is a Markov probability meamire, on the space 7 of infinite Young tableaux, with transition probabilities
n
where n = IXI is the number of squares of A. Using the hook-length formula, we cau rewrite this expression in the form
34. A q-ANALOGUE O F T H E HOOK WALK ALGORITHM
127
Let us explain our notation. Let M be a point in a diagram X (regarded as a part of R): with coordinates i , j . The number c(M) = j - i is called the contents of M . Denote by MI, M2, ...,Md the successive vertices of the internal corners of A, and by Nl, N2, ...,Nd-1, the vertices of its external corners lying between them. In (3.4.2), xk = c(Mk), yk = c(Nk) are the contents of the vertices of the south-east border of X (see Figure 19). We write Rk in place of RAAif the square that distinguishes A from X is attached to Mk. In what follows, we will need to inscribe X into the rectangle with vertices No, Nd; denote by yo, yd their contents.
FIGURE19. For the definition of cotransitional probabilities. It follows from the well-known formula (n
X
+ I )f x = C A r x ffAAthat
Let be the complement of X in the rectangle (Figure 19). Using the hooklength formula, we can write the recurrence relation (3.4.1) for the diagrams A /" in a form similar to (3.4.3):
X
- yi-l)(yj - x j ) . Conversely, in order to prove the hookwhere S = xl,i<j
128
3.
T H E PLANCHEREL MEASURE O F THE SYMMETRIC GROUP
In the same paper a q-analogue of (3.4.3) is stated. The main result of [102] is a construction of a Markov process such that the final probabilities of states of this process coincide with the summands in (3.4.4). In the modification of this algorithm suggested in [103],the final probabilities coincide with the summands in (3.4.3). 4.2. Random knot tableaux. Consider a Markov measure P in the space of Young tableaux 7 with transition probabilities
where t = (XI, X2, . . . , A,, . . . ). Following [17],we call the measure P central if for each four-tuple of Young diagrams such that X /" p /" A, X /" v /" A we have
It may be seen from the definition that for a central measure the probability
does not depend on p1,. . . , pn-l. Ergodic central probability measures correspond to characters of the countable symmetric group B,, or the infinite-dimensional Hecke algebra H,(q) with q > 0 (see [17], [19]). The following description of these measures is well-known. The ergodic central measures are indexed by pairs of nonincreasing sequences a = {ai)gl, /3 = {Pi)zl of nonnegative numbers satisfying the condition
The transition probabilities of the measure corresponding to a pair a,P are given by the formula
where the extended Schur functions s x ( a ;P) are defined, for example, by F'robenius' formula ([49], 1.7.6)
xi
+
>
(-l)n+l /?: for all n 2. The where p l ( a ; p) = 1 and p,(a;P) = Cia: Plancherel measure corresponds to the zero sequences. Expressions of the form W(r, q) = (1 - qT+l)/(l q ) ( l - qT)play an important role in the description of irreducible representations of Hecke algebras (see [33]). Given a triple of diagrams p /" X /" A, let N, and MI, be the vertices to which the squares X\p and A\X, respectively, are attached. The number r p x A = MI,) c(N,) = xk - ,y is called the axzal distance.
+
$4. A q-ANALOGUE OF T H E H O O K WALK A L G O R I T H M
129
DEFINITION.A central measure P is said to be a knot measure if there exists a number q such that the sun1
does not depend on the pair p
/" A.
The choice of the term is due to the fact that the characters of H , ( q ) associated with knot nleasures are used for constructing topological invariants of links in IR3. See [19]for a simple description of knot ergodic central measures. Let us enumerate the parameters of these measures using the notation [n] = ( I - q n ) / ( l- q ) .
EXAMPLE 1 . Let a
= ( ( 1 - q ) q k } E 0 ,P = 0 .
EXAMPLE 2. Asslime that
0
= 0, P =
In this case
( ( 1 - q)qk)r=,. Then
EXAMPLE 3. Let 0 < q < 1, 0 = { q k ( l - q ) / ( l - q r n ) } T ~Pt ,= 0 .
EXAMPLE 4. Let
0 =
0,
=
{ q k ( l - q ) / ( l - q r n ) } ~ In ! ~this . case
x,?,
By definition ( [ 4 9 ] ,1.1.5),n(p) = (i - l ) p i for a diagram ( p l , p 2 , . . . ). The first two examples are special cases (when t = 0 and t = 1) of the following twoparameter family of ergodic knot central distributions.
EXAMPLE 5. Asslime that 0
1, 0
0
< t < 1 . In this case
=
((1
-
t ) ( l - q ) q k } p = oP, = { t ( l - q)qk}r==,for
4.3. Transition probabilities. Let X be a Young diagram whose border has vertices at points M I , N l , . . . , N d P l , M d , and let xl < yl < . . . < yd-1 < xd be the conterits of these points. If X /" A and the new square in A is atta,ched to the kth vertex M k , we write R k , . . . , ck in place of R x A , . . , cxi\ in the notation for trarlsitiorl probabilities.
3.
130
T H E PLANCHEREL MEASURE OF T H E SYMMETRIC GROUP
PROPOSITION. The transition probabilities of the knot central distributions from Examples 1-4 i n Section 4.2 can be written i n the form
For the family from Example 5, the transition probability is
PROOF.The transition probabilities are the quotients (3.4.11). Formulas (3.4.19)-(3.4.23) are obtained by substituting (3.4.14)-(3.4.18) in (3.4.11) and using the following identities. Let (u, v) be the coordinates of M k . Then n ( A ) - n(X)= u = x kd+ l ( x i - yi-1) and
which implies (3.4.19) and (3.4.20). Furthermore, the contents c(b) of a square b does not depend on the surrounding diagram, so that, for instance,
since c ( a ) = c ( M k )= xk for the new square a
= A\X
and m = -yo.
Observe that
(3.4.24)
C k ( 9 ) = qxkRk(q)r C A ' A ' ( ~=) R A A ( ~ )C> A ) A ' ( ~ )= T A A ( ~ ) . $5. The q-analogue o f t h e h o o k - l e n g t h f o r m u l a
5.1. The q-hook-length f o r m u l a . Let y be the Young graph described in the Introduction. Let us assign to each edge ( v ,p ) , v /" /I, a multiplicity m,,(q) depending on a parameter q by setting
where N = 1p1, c(b) is the contents of b E p, and a = p\v is the square that should be removed from p t o obtain v . Denote by y ( q ) the graph y with the system
$ 5 . T H E q-ANALOGUE O F T H E HOOK-LENGTH FORMULA
131
-
of edge multiplicities {m,,(q)); since m,,(3.4.1) = 1, one can think of y ( q ) as a deformation of the ordinary Young graph. We define recursively a q-analogue f,(q) of the dimension f, by setting fm(q) 1 for the empty diagram p = 0 and
where v ranges over Young diagrams that immediately precede p in y . Clearly, f,(3.4.1) = f,. In Section 5.6 below we will prove
THEOREM 1. The q-dimension f,(q) can be computed by the q-hook length formula,
where N = )pi, [k] = 1
+ q + . . . + qk-l,
and [N]!= [1][2].. . [ N ]
We emphasize that the q-dimensions f,(q) are polynomials, though the multiplicities m,,(q) are rational functions in q. For instance, m,,(q) = q(1 q2)/(1 q) for v = (21) and p = (22).
+
+
5.2. Cotransition probabilities. Denote by g,(q) the right-hand side of the hook-length formula (3.5.3). In order to prove Theorem 1 it suffices to check that the functions g,(q) satisfy, like f,(q), the recurrence relation (3.5.2), that is, C, p,,(q) = 1 for all p # 0, where
Let p be the (inverted) Young diagram that is the complement of a diagram X in the rectangle (Figure 19, in Section 4.1), and let yo < x l < yl < . . . < yd-1 < xd < yd be the parameters introduced in Section 4.2. Let us express the summands pk(q) = p,,(q) in terms of these parameters, assuming that the square a = p\v is attached t o the point Mk. Following [17], we may call p,,(q) the cotransition probabilities (for central measures of the graph y ( q ) ) .
where S(q) =
d
qYz - ~
t qxl=- qYofYd. ~
PROOF. Let ud, ud-1,. . . ,uo and vo, vl, . . . , vd be the row and column numbers, respectively, of the points No, Nl, . . . , N d Thus xk = vk-1 - ud-k and yk = vk - ud-k for k = 0 , 1 , . . . , d. We also set zji = vj-1 - ud-i; then zkk = xk, zk+l,k = yk, and zd+l,o = vd - ud = yo yd. The expression for S(q) can be rewritten in the form
+
3.
132
T H E P L A N C H E R E L MEASURE O F T H E S Y M M E T R I C G R O U P
PROOF. It is clear from the definition that n(p) - n(u) = ud - ud-k - 1 (this is the row number of the square a = p\u, in the diagram p, decreased by one). Therefore
where
n,
n
[1+ hicI i=l [hzcl
n
k-l
r-1
=
=
[xk - YO]
[xk - yi] i= l Xk - xz] '
Here r, c are, respectively, the row and column numbers of the square a the diagram p. The lemma follows now from (3.4.19), since
=
p\v in
and ud = -yo.
PROOF.Let a square b E p lie in the uth row and vth column of the rectangle, and denote by r, c its row and column numbers in the diagram p. If c(b) = c - r is the contents of b in p and E(b) = v - u, then c(b) E(b) = yo yd = const, since c + v = yd 1 and r u = -yo 1. We can rewrite (3.5.1) in the form
+
Let
+
+
+
n(i,j ) = {b E p : ud-i < u < ud-i+l, uj-1 1 u < vj}.
+
Then
comparing with (3.5.6) yields
Since E(a) = xk, the lemma follows. The proposition follows immediately from the lemmas.
$ 5 . T H E q-ANALOGUE O F T H E HOOK-LENGTH FORMULA
133
5.3. Algebraic identities. As shown in Section 5.1, Theorem 1 means that for each q > 0 the sum of the numbers (3.5.5) is equal t o one. Since the expressions (3.4.19)-(3.4.23) are interpreted as probabilities, their sums are also equal to one. The corresponding algebraic identities are valid for arbitrary (not necessarily integral) parameters yo, xl , . . . , xd, yd for which they make sense.
THEOREM 2. Let {xk);+, be pairwise distinct real numbers, {Yk)%=oarbitrary d real numbers, z = C k = l xk - C::: yk, and q > 0. Then
- where S(q) = Cl
-
q-"d-*)(qvj-l - q"'~).
PROOF. The parameters {xk), {yk) associated with a Young diagram X are integers with
and z = vo - uo = 0. Under these conditions, formulas (3.5.7)-(3.5.10) are already proved in Section 4.2. Below we use the hook walk algorithm to give another probabilistic interpretation of identities (3.5.7)-(3.5.11) with parameters satisfying (3.5.12). Translating the parameters by a constant leaves the summands unchanged, which allows us t o drop the condition z = 0. Since the left-hand sides are rational functions in qYk, qxk, the ordering (3.5.12) and the condition that {xk), {yk) are integral are also irrelevant. For q = 1 we recover formulas (3.4.3)-(3.4.6) from Section 4.1. 5.4. Transition probabilities. In this section we will describe the Markov chain which is a q-analogue of one of the versions of the hook walk algorithm from [102], [103](cf. also [146]). Given two sequences of real numbers (uo < u1 < . . . < ud) and (vo < vl < . . . < vd), consider the rectangle with vertices (uo,vo), (uolvd), (ud,ud), (ud,vo) (in Figure 19 the axis u goes downward, and the axis v goes t o the right). Given a point T with coordinates (u, v), denote by c(T) = v - u its contents. Denote by M k ,Nk the points with coordinates ( u ~ - vkPl), ~, (udPklvk); then xk = c(Mk), yk = c(Nk). Possible states of the hook walk belong to the set X = {(i,j ) E Z: 1 i 5 j 5 d). It is convenient t o think of a state (i, j) E X as the point T = ( U ~ - ~ , Vin~ - ~ )
<
3.
134
T H E PLANCHEREL MEASURE O F T H E SYMMETRIC G R O U P
the rectangle. The hook at this point is the union H ( i ,j ) = A U L of two intervals: v v j - l ) which joins T with M j , and the the hook arm A = { ( u d P jv, ) : ui-1 hook leg L = { ( u ,ui-1): u d - j u u d - i ) which joins T with hfz.The hook length is equal to x j - xi. Note that a point Q E H ( i , j ) is uniquely determined by its
< <
< <
contents h = c ( Q ) . The transition probabilities of the Markov chain being defined depend on a parameter q > 0. Given a current state ( i , j ) E X, pick a point Q E H ( i ,j ) at random with density Cqhdh, where C is a normalization constant (for q = 1 this exponential distribution reduces to the uniform one). Constrlict a new state ( i l ,j l ) depending on Q . There are two possibilities (Figure 20): ( A ) if Q E A and urn-1 < v < u,,, set ( i l , j l )= ( i , m ) , ( L ) if Q E L and ud-,, < u < u d P m + l , then ( i t ,j l ) = ( m ,j ) .
FIGURE20. Hook walk algorithm. Note that the difference ( j - i ) with i < j decreases with probability one. After at most d steps the hook walk will reach the subset X , = { ( i ,j ) E X : i = j ) and the final state ( k ,k ) will stop changing. We will show that the numbers R k ( q ) , . . . , pk(q) coincide with the probabilities of final states in the hook walk for appropriate initial distributions. Let 11s proceed to the description of these distributions.
5.5. Initial distributions. Associate with a state ( i ,j) E X the rectangle II(i,j ) = { ( u , ~ )ud-i : < u < u d p z + l , utPl < v < ut}, and take its q-area
as the statistical weight of this state. Each subset Y Py on Y with probabilities
cX
gives rise to a distribution
where S y = C(i,j)EY S i j ( q ) . The following examples of subsets are of interest: a ) a vertex { ( l , d ) } ; b) Y = { ( i , d ) :1 i d } ;
< < < d);
Y = ( ( 1 ,j ) : 1 < j d ) the entire set X . C)
T H E O R E3M . Let R k ( q ) , r k ( q ) , ck ( q ) ,pk ( q ) be the probabilities of a final state ( k ,k ) E X , i n the hook walk with initial distributions (3.5.14) supported by the sets from Examples a)-d), respectively. Then formulas (3.4.19)-(3.4.22) and (3.5.5) hold.
55. T H E q-ANALOGUE O F T H E HOOK-LENGTH FORMULA
135
This theorem implies identity (3.5.11) and Theorem 1. It also provides an independent proof of formulas (3.5.7)-(3.5.10) and an efficient way to simulate random knot tableaux. 5.6. Proof of Theorem 3. Denote by R P ) ( q ) , r P ) ( q ) , c P ) ( q ) , p F ) ( q ) the probabilities that the hook walk with initial distribution (3.5.14) supported by = {(i, j)), YJi') = {(s, j): i s j), yJii)= {(i,t): i t j), the sets
YP'
and Y;")
= {(s,t): i
Z"23 -21.
-
2-1
< s < t < j),
< <
< <
respectively, stops a t ( I c , Ic) E X,.
Let
ud-j.
LEMMA3. The following recurrence relations hold:
PROOF. The coefficients of these relations coincide with the transition probabilities at the first step of the walk, and the formulas themselves are various versions of the formula of total probability. For example, (q". - qzi')/(qxi - q " ~ )is the probability (the initial state being T = (i, j ) ) that the chosen point Q lies in the leg L of the hook H ( i ,j ) , and (qZt3- qX3)/(q"%- q " ~ )is the probability that Q E A.
PROOF. First of all, observe that (3.5.20)-(3.5.22) can be rewritten in the form
Here we use the relations yi
xi
- ~ i + l= , ~
-
z23. . and yj-1 - zi,j-l = x j
-
z23.
136
3.
T H E PLANCHEREL MEASURE O F T H E SYMMETRIC G R O U P
Lemma 4 follows easily from (3.5.23)-(3.5.25) and Lenlma 3 by induction on ( j - 2 ) . For example, we can use (3.5.25) and (3.5.20) to write the right-hand side of (3.5.18) in the form
which implies (3.5.22). Theorem 3 follows immediately from Lemma 4, since R k ( q ) = R,( 1 4 ( q ) , . . . ,
~ k ( q=) p r d ' ( q ) . 56. Multiple Selberg integrals 6.1. Versions of Selberg integrals. In an old paper [ 1 5 6 ] ,A. Selberg computed the integral
As indicated in [ 6 9 ] ,recently (1987) he found another multiple integral, which is, however, taken over the simplex rather than the cube:
In this section we conlpute an integral similar t o (3.6.2):
First of all, let 11s establish the relations between the integrals (3.6.1)-(3.6.3).
PROPOSITION. The integrals (3.6.2) and (3.6.3) are equivalent; both are consequences of (3.6.1).
56. MULTIPLE SELBERG INTEGRALS
137
PROOF.The implication (3.6.1) + (3.6.3) was communicated t o me by G. I. Olshansky. His arguments are as follows. Substitute t j = r j / B in the integral (3.6.1); then
Iv(~)\~'
=
n
Iv(T)
12' . B ( ~ - ' ) ~ ' tp-'
Multiply both sides of (3.6.1) by B account the formula
~
dt,
~and (let B ~
~
+- co.~ Taking ) ~into ~
we obtain
and both sides of (3.6.1) have limits. We arrive at the following formula:
/ .. / n .
RI;
(3.6.4)
/ti - tilZeexp
n
( - Ctj) t 4 - I d t j j=1
i<j
j=1
" r ( A + (m - j ) O ) r ( j 0 + 1)
=n j=l
r(e+ 1)
Now make another change of variables:
so that
CEl xj
=
1. The Jacobian of this change is equal to
sm-I,
so that
The integral (3.6.4) takes the form
The first factor in the right-hand side coincides with the classical definition of the gamma function, (3.6.8)
I"
SmA+(m-l)m8-le P Sds =
r(mA
+ (m
-
l)me),
and we arrive at (3.6.3). In other words, (3.6.4) and (3.6.3) are equivalent.
138
3.
T H E PLANCHEREL MEASURE O F T H E SYMMETRIC G R O U P
We now prove the equivalence of (3.6.2) and (3.6.3). As in the first part of the proof, make the change of variables (3.6.5) in the integral (3.6.2). The simplex t l . . . tm 1 goes to the direct product of the interval [O, 11 and the simplex
+ +
<
The integral (3.6.2) factors into the product
It remains to observe that the first factor in the right-hand side is the classical beta integral
The equivalence of (3.6.2) and (3.6.3) is proved. 6.2. Mul.tiplicative Markov chains on the Young graph. Our idea of computing the integral (3.6.3) is to present certain central measures on the Young graph and its deformations for which formula (3.6.3) becomes the generalized Poisson representation of the associated harmonic function. The representing measure p on the boundary of the graph will be obtained as the radial limit of the distributions Mn of a Markov chain at the nth level y, of the Young graph. In this sense one may say that we compute the integral by passing explicitly to the limit in appropriate integral sums. The fact that for our distributions Mn it is really possible (and even easy) to pass to the limit is due to their multiplicativity: the weight Mn(X) of a Young diagram X E yn is the product of simple similar factors over all squares of A. In order to make the main idea more precise, let us first consider the special case 6' = 1 of the integral (3.6.3). Denote by y ( m ) the truncated Young graph. By definition, it consists of Young diagrams X E y with at most m rows, the edges being the same as in the complete Young graph. It is convenient to interpret y ( m ) as the branching graph of Schur functions sx in m independent variables:
Thus this graph is multiplicative, and its boundary E ( y ( m ) ) is easy to compute. It is homeomorphic t o the simplex
56. MULTIPLE SELBERG INTEGRALS
a vector
ai E
139
A, being associated with the nonnegative harmonic function
The most interesting example is the ergodic Markov chain on y ( m ) corresponding to the frequencies a = ( l l m , . . . , l l m ) E A, (one may say that this is the truncated analogue of the Plancherel measure). The transition probabilities of this chain are defined by the formula
where
are the transition probabilities of the true Plancherel measure. The distribution of the state a t the nth level y,(m) of the graph y ( m ) is given by
where c(b) = j - i is the contents of the square b = (i,j) of X E y n ( m ) , and h(b) = Xi A'3. - j - i 1 is the hook length of this square. There is an interesting one-parameter deformation of the distributions (3.6.14) which is still central but no longer ergodic. The transition probabilities, which depend on a parameter v > m - 1, are defined by the formula
+
+
These probabilities are well-defined, as may be seen by computing the moments of the transition probabilities of the Plancherel measure, see (4.1.18):
The Markov chain with transition probabilities (3.6.15) is central, since the probabilities of all Young tableaux of shape X coincide and are equal t o M ~ ) ( x/ )d(X), where
The original formulas (3.6.12) and (3.6.14) are obtained from (3.6.15) and (3.6.17) by letting v + m.
140
3.
T H E PLANCHEREL MEASURE O F T H E SYMMETRIC GROUP
Let us now find the Poisson integral representation for the central measures (3.6.15) on y(m). Recall that the radial embedding of the graph y(m) into its boundary A, is defined by the formula
PROPOSITION. Consider a sequence of diagrams A(") E limits
yn(m) such that the
exist. Then there also exists the limit
where
In other words, the weak limit of the discrete distributions i ( M P ) ) on Am exists and equals the absolutely continuous measure
PROOF.Using the well-known formula
1
nm
bEX
+
n i < , ( X i - Aj j - i ) = nj=,(xi+m-i)! ,
rewrite (3.6.17) in the form
n!
n
m
++
m
+
( m- i l ) x , M?) ( A ) = ( X i - ~ ~ + j - i ) ~ (xv n- i l ) ~ , ( ~ im - i ) ! +m-i)!. i=l l
we can write
xJI
56. MULTIPLE SELBERG INTEGRALS
In this computation we have used the asymptotic formula
The proposition is proved.
Since (3.6.22) is a probability measure, we have obtained the speCOROLLARY. cial case 8 = 1 of Selberg's integral (3.6.11). More generally, writing the generalized Poisson integral for the central Markov chain with probabilities (3.6.15), we obtain the formula
6.3. B r a n c h i n g of J a c k p o l y n o m i a l s , a n d S e l b e r g ' s i n t e g r a l . In this section we derive the general integral (3.6.3) along the same lines as the special case 8 = 1 in the previous section. However, we must modify the edge multiplicities of the truncated Young graph y(m) by introducing a parameter 8 . For this purpose we use the Jack symmetric polynomials (see [ I l l , 1 1 2 , 1 0 6 , 1591). Jack polynomials P x ( x ;8 ) can be defined as the limiting cases of Macdonald polynomials P x ( x ;q, t ) for q = I E , t = 1 OE, and E -+ 0. Thus, in the general definition of Chapter 2, $7, we should modify only the bilinear form (2.3.1). Namely, we should set
+
+
where l ( A ) is the number of nonzero rows of a Young diagram A = ( I r ' , 2 Q , . . . ) and z~ = 1'1 r l ! 2'2 r2! . . . . Note that the Jack polynomials J x ( x , 8 ) in [159] have another normalization:
(3.6.27)
8' J A( x ;8 ) = PA( x ;8 )
n
(a@)
+ ( 1( b )+ 1 ) 8 ) ,
bEA
where a ( b ) = Xi - j and l(b) = XI, - i are the arm and leg lengths of the hook a t the square b = (2, j ) E A, respectively. For 8 = 1 the Jack polynomials coincide with the Schur functions, P x ( x ; 1 ) = s x ( x ) , and for 8 = 0 they reduce to the monomial symmetric functions, P x ( x ;0 ) = mx( x ) . Pieri's formula for Jack polynomials is
where the multiplicities are defined by
(3.6.29)
xs(A,A )
nyeT
=
n( ver
a(b) a(b)
+ ( l ( b )+ 2 ) 8 ) (
+ (l(b)+ 1 ) 8
+ + + + +
a ( b ) 1 1(b)8 a(b) 1 ( l ( b ) 1 ) 8
The product is over all squares of the column of A strictly above the new square b = A \ A. Several first levels of the branching graph of Jack polynomials were shown in Figure 10 on page 43.
142
3.
THE PLANCHEREL MEASURE O F T H E SYMMETRIC GROUP
The dimensions of the Jack graph can be computed by a hook-length formula which generalizes the ordinary formula from [92]:
This formula was obtained by Stanley [159]; for 19 = 1 it reduces to the hook-length formula for the dimensions of the Young graph. The boundary of the Jack graph J(') does not depend on the parameter Q > 0 and can be identified with the simplex (3.6.27) from Section 6.2. On the contrary, the Poisson kernel does depend on 8 and coincides with Jack polynomials: (3.6.31)
@(A;a ) = Px(a; Q), A E y ( m ) , a E A,.
Following the scheme of the previous section, we must now define an analogue of the transition probabilities (3.6.15) which depends on 8 and determines central Markov chains on the Jack graph ~ ( ' 1 . Remarkably, this can be done by using the following simple idea, which also provides a significant interpretation of the formal parameter 8. Let us consider, along with a Young diagram X E y, its scaled versions X(Q) obtained by preserving all horizontal dimensions (row lengths) and multiplying all vertical dimensions (column lengths) by Q. For example, if X = (72, 44, 12) and Q = 112, then X(1/2) = (7, 42, 1). In the general case, the rectangular diagram X(Q) is no longer a Young diagram; however, its minima xk(Q)and maxima yk(Q)are in a one-to-one correspondence with the minima and maxima xk, yk of the original Young diagram A. Notice that the contents of a square (or, better to say, of a point) with coordinates i, j in the scaled diagram X(Q) should be defined by
For 8 = 1 we recover the ordinary contents. The contents of the external and internal corner squares of the scaled diagram X(Q) will be denoted by xk(Q)and yk (Q),respectively. Let us define the transition probabilities by the same formula (3.6.15), but applied not to a Young diagram X but rather to its scaled version X(Q). This is possible, since due to Theorem 32 we have the definition of transition probabilities for any rectangular diagrams. So set
(these are the Plancherel transition probabilities for the scaled diagram X(Q))and let
where k = 1 , 2 , . . . , d runs over the numbers of external corners of A. It follows immediately from the general lemma, which claims that the first three moments of the transition distribution do not depend on the shape of the rectangular diagram, that (3.6.34) defines a probability distribution. Using formula (3.6.29) for the edge multiplicities of the graph ~ ( ' 1 , one can easily verify that the hlarkov chain with transition probabilities (3.6.34) is central
56. M U L T I P L E S E L B E R G I N T E G R A L S
143
on the corresponding truncated graph ~ ( ' ) ( m )The . distribution of the state at the nth leiel Jn(0) ( m )is given by
where ( x ) , = x ( x The limit as n
+ 1 ) . . . ( x + n - 1 ) is the Pochhammer symbol. -t
oo can be obtained as in (3.6.20), above.
PROPOSITION. Let v = A + ( m - 1)d, A > 0 , and assume that for a sequence of Young diagrams the limits (3.6.35)
lim
n-cc
A:"' n
j = 1 , 2 , . . . , m,
-=aj,
exist. Then lim n m l M ; ~ ' * ) ( A ( ~ =)C) m ( 0 )
(3.6.36)
n-cc
n
n m
lai - a j '1
l
,
j=1
where the normalization constant C m ( 0 ) is
COROLLARY. The limiting measure with density (3.6.36) is a probability measure. Thus we have obtained Selberg's integrals (3.6.2) and (3.6.3). Writing the generalized Poisson integral for the central Markov chain with transition probabilities (3.6.34) on the Jack graph $') ( m ), we obtain the identity
m times
-
PA(^ ,... , 1 ; 6 )
r(n+ m A + m(m
-
rI r ( A j + A + r((m0 +j1) )0 ) r ( j 0 + 1 ) -
1)0) i=l
CHAPTER 4
Young Diagrams in Problems of Analysis 51. Rectangular diagrams and rational fractions In this section we extend the set of Young diagrams y t o the set of rectangular diagrams Do. The motivation for this extension is that rectangular diagrams provide a very convenient tool for graphical representation of pairs of interlacing sequences of real numbers. Looking ahead, we mention that in the Markov moment problem rectangular diagrams correspond t o canonical measures (the term used by Krein and Nudelman) . It is important in what follows that the Plancherel transition probabilities, which were originally defined only for Young diagrams on the basis of the representation theory of symmetric groups, can be as naturally defined for any rectangular diagrams. We show that the moments of transition distributions are given by the extended symmetric functions introduced earlier in the statement of Thoma's theorem. 1.1. Rectangular diagrams and interlacing sequences. Let v = w(u) be a continuous piecewise linear function (Figure 21) such that (i) wl(u) = &I, and (ii) W ( U ) = u - zI for some z E R and sufficiently large Iu~.
FIGURE 21. A rectangular diagram. Such functions will be called rectangular diagrams, and the number z = z(w) will be called the centre of the diagram w. The subgraph
146
4.
YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS
translated by z and rotated by 45" (see Figure 7 on p. 33) resembles a Young diagram; this fact, which explains the choice of the term, will be used in what follows. By definition, the area of the diagram is the area A = A(w) of its subgraph (4.1.1). We will use rectangular diagrams to obtain a visual representation of the relative position of a pair of interlacing sequences. Given a diagram w, denote by 21,. . . ,xn (respectively, yl, . . . , yn-1) the points of local minima (maxima) of w. These sequences interlace:
The centre of the rectangular diagram with extrema (4.1.2) is located a t the point z = C xk - C yk, and its area equals
(the unit of area is assumed to be the area of the square { ) u (5 v < 2 - 1 ~ 1 ) ) . Conversely, for each pair of interlacing sequences (4.1.2) there exists a unique rectangular diagram w whose extrema coincide with these sequences. Indeed, take the point
as the centre, and recursively define the values of w at the points of extrema by setting w(xl) = z - X I and
for k = 1,.. . , n - 1. Then w(xn) = xn - z , and both conditions (i), (ii) in the definition of a rectangular diagram are satisfied. The uniqueness of the diagram with given extrema is guaranteed by (ii). Besides being visual, the representation of interlacing sequences by diagrams has another, more significant, advantage: we can define in an obvious way the limiting diagrams which characterize the asymptotics of mutual separation. We will return to this subject in $$5 and 7. For the rest of this section, we fix an interval [a, b] and denote by Do = Do[a,bj the set of diagrams with w(u) = (u- z ( for u 6 [a, b]. True Young diagrams are a special case of rectangular diagrams, so that y c Do. Besides conditions (i), (ii) they are characterized by the following additional requirements: (iii) all points of extrema {xk), {yk) are integral, and (iv) the centre is located a t the origin, z = 0.
1.2. The transition measure of a rectangular diagram. Consider a rectangular diagram w E Do with interlacing extrema
and let p be the measure with atoms of (positive) weights
§I. RECTANGULAR DIAGRAMS AND RATIONAL FRACTIONS
147
at the points xk, k = 1,. . . ,n. These numbers coincide with the coefficients of the partial fraction expansion
Multiplying both sides by u and letting u at a discrete probability distribution.
+
cc yields
C pk
=
1. Thus we arrive
DEFINITION. The probability measure p with atoms of weights (4.1.7) at the points xk, k = 1 , . . . ,n, will be called the transition distribution of the rectangular diagram w . It is important to emphasize that condition (4.1.6) that the zeros and poles of the rational fraction (4.1.8) interlace is necessary and sufficient for the coefficients pk to be positive.
Formula (4.1.8) establishes a one-to-one correspondence bePROPOSITION. tween the set Do of rectangular diagrams (or, which is the same, the set of interlacing sequences (4.1.6)) and the set Mo of finitely supported discrete probability distributions on the real line. Given a distribution {pk, ~ k ) ; = ~the , second sequence {yk) can be recovered as the set of roots of the polynomial
Since Young diagrams are a special case of rectangular diagrams, we may apply to them the above definition of the transition measure. In Section 1.4 we will see that it is equivalent to the definition of the transition distribution of the Plancherel growth process, which motivates our using the term in a wider context. In Section 3.2 we will show that the bijection between the subsets Do and Mo established in (4.1.7) and (4.1.9) can be extended by continuity to a homeomorphism of the completed spaces D[a, b] and M [ a ,b]. For this purpose we will need the moments of transition distributions. 1.3. Moments as symmetric functions. Denote by hn the ordinary moments of a discrete probability distribution {pk,x ~ ) $ = ~ :
In a similar manner, we introduce the moments of a rectangular diagram with extrema { ~ k ) $ , ~{yk):,: , by
The moment generating functions are
148
4.
YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
and
It follows form the defining formula (4.1.8) that these functions are related by the formula
which can also be rewritten in the form
Formula (4.1.15) is one of the fundamental identities of the theory of symmetric functions (see [49]). In this context
are the complete homogeneous symmetric functions, and
are the Newton power sums. The generating function for the polynomials hn is
Each symmetric polynomial in variables x = { x i ) can be naturally extended t o a polynomial in two sequences of variables x = { x i ) , y = { y i ) . The resulting extended symmetric functions turn out t o be useful in various situations (A-rings, characters of the infinite symmetric group B,, see [17]),in particular, in connection with rectangular diagrams. Recall the definition of these functions. The extended power sums are defined by
and the extended complete homogeneous symmetric functions are expressed in a standard way in terms of p,:
where p = (1'1, 2'2,. . . ) ranges over partitions of the number n = generating function for these polynomials equals
Comparing this with formula (4.1.14) rewritten in the form
xk,lk r b The -
51.
RECTANGULAR DIAGRAMS AND RATIONAL FRACTIONS
149
yields the following result.
Let w E Do be the rectangular diagram constructed from interPROPOSITION. lacing sequences x = {xk}, y = { y k ) , and let pw be the corresponding transition distribution. Then their moments are extended symmetric polynomials in x , -y:
For example, hl = pl = C xi - C yi = z is the centre of the diagram w and the mean value of the measure p , and h2 - h: = (p2 - p:)/2 = A is the area of the diagram and the variance of the measure.
1.4. Relation to the Plancherel growth of Young diagrams. The Plancherel measure M of the infinite symmetric group 6, was defined in $1 of Chapter 3 as the Markov chain on the Young graph y with transition probabilities PXA =
dim A (N+l)dimX1
XEYN,
X/"A.
Let us write pk in place of p x ~ if the square that distinguishes A from X is attached to the minimum xk of A. PROPOSITION.The weights of the transition distribution of the Plancherel measure at the Young diagram X are equal to
Thus this transition distribution coincides with the transition measure of the diagram X in the sense of definition (4.1.8). The proof is based on the hook-length formula, which implies that
Let us compute this quotient. Assume that the new square is located in the r t h row and cth column of the diagram A. Since the squares outside the r t h row or cth column have equal hook lengths in the diagrams X and A, we can rewrite (4.1.21) in the form
The factor
n hTi/(hTi+ 1) factors into products of the form
whence
Similarly,
150
4.
YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS
So for Young diagrams the definition (4.1.8) in terms of rational fractions is exactly equivalent to the group-theoretic definition of the Plancherel transition probabilities (4.1.19). 52. Continuous diagrams and R-functions The purpose of this section is to complete the space of rectangular diagrams Do by their uniform limits, called continuous diagrams. We show that the definition of the transition distribution of a rectangular diagram can be extended by continuity to arbitrary continuous diagrams. The bijection established in 51 between the sets Do and M o becomes a part of the bijective correspondence between the space of all diagrams V and the space M of probability distributions on the real line. To establish this correspondence, we use the moment method. 2.1. Definition of continuous diagrams. Following [I181 (see also [12]), by diagrams we mean continuous functions v = w(u) satisfying the following two properties (Figure 22) : (i) Iw(u1) - w(uz)I I 1% - 2121, and (ii) w(u) = lu - zl for some z E IR and sufficiently large lul. The number z = z(w) from (ii) is'called the centre of the diagram w; the area A = A(w) of the diagram w is defined as the area of the region
that is,
(the unity of area is assumed to be the area of the square {IuJ 5 v
< 2 - lul}).
FIGURE 22. A continuous diagram. Fix an interval [a,b] and denote by V = V[a, b] the set of diagrams with w(u) = Iu - zI for u @ [a,b]. Endow the space V with the uniform convergence topology. Note that if we do not wish to fix an interval [a,b] beforehand, then the definition should be modified. By definition, a sequence of diagrams converges, w, + w, if w,(u) + w(u) uniformly in u E IR, and there exists c > 0 such that w,(u) = lul for lul > c for all n = 1 , 2 , .. . .
$2. CONTINUOUS DIAGRAMS AND R-FUNCTIONS
151
A diagram w is called rectangular if it is piecewise linear and wl(u) = f1 for (almost) all u E W (see Figure 21, on p. 145). A rectangular diagram is uniquely determined by the coordinates of its minima X I , .. . , x, and maxima yl, . . . ,y,-1, which form a pair of interlacing sequences:
Conversely, each pair of interlacing sequences (4.2.3) uniquely determines the rectangular diagram w with the corresponding extrema and with centre a t the point z = C xk - C yk. The area of the rectangular diagram is
Note that the set Do of rectangular diagrams is dense in V. 2.2. The R-function of a diagram. Given a diagram w E V[a, b], consider the function a(u) = i(w(u) - Iuj). This function enjoys the following properties:
0'(u)
+ >
~ ' ( u ) 1 0 if wl(u) - 1 5 0 if max a ( u ) = w(0)/2,
=-
u u
< 0, > 0,
u
The differential da(u) = ul(u) du determines an absolutely continuous (signed) measure called the charge of the diagram; it is supported by the interval [a, b], and the total variation of the charge equals Jdal = w(0).
DEFINITION. The R-function of a diagram w E V[a, b] is given by (4.2.4)
2
1 Ru(u) = ;exp[
= - exp
u
:J,
-
d(w(t) - Itl) t-u .
This function is defined and holomorphic outside the interval [a, b].
EXAMPLE 1. Let w = 0 , where
then Ra(u) = (U - d-)/2
for u
>. 2.
EXAMPLE 2. Let w = w, be the diagram which is centered a t z < 1 and q = 1 - p) and linear in the interval [-I, 11, that is,
0
(4.2.6)
1-uz wP(u) =
if
(~121,
if
1u1 >. 1.
Then da(t) = and R,(u) = (u
{qdu -pdu
if
-l
if
O
+ 1)-q(u - 1)-p for u 2 1.
=p -q
(where
152
YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS
4.
EXAMPLE 3. Let w be the rectangular diagram, with centre z , constructed from a pair of interlacing sequences (4.2.3). For each smooth function f the following version of the Newton-Leibniz formula holds:
Indeed, it follows from X I < 0 < x, that for some k either yk-1 xk 5 0 5 yk. In the first case both sides of (4.2.7) are equal to
c l;
k-1
f ' ( t )dt
i=l
-
LXk 2 f l(t)dt -
k+l
JX.
< 0 5 x k , or
f l ( t )dt;
Yt-1
the second case is analogous. Let us find the R-function of a rectangular diagram. LEMMA.
Let a rectangular diagram be constructed from the roots of polynomials
Then R ( u ) = Q ( u ) / P ( u ) . PROOF.
It suffices to apply (4.2.7) t o the function f ( t ) = log(u - t ) .
2.3. The transition measure of a continuous diagram. Denote by M = M [ a ,b] the space of probability measures on the interval [a,b] endowed with the weak topology, and let M o c M be the dense subset of finitely supported measures. DEFINITION.
The R-function of a measure p E M [ a ,b] is its Cauchy-Stieltjes
transform
Like the R-function of a diagram, it is defined and holomorphic outside the interval [a,bl. EXAMPLE. The R-function of a discrete measure p E M o with atoms of weights pk a t points x k , k = 1 , . . . , n, equals m
DEFINITION. A probability measure p is called the transition measure of a diagram w if their R-functions coincide:
R,(u) =
J/:
d ~ ( t ) - exp - J b d ( w ( t ) - I t ( ) u-t u 2 , t-U
-=
= Rw(U).
THEOREM [118, 471. For each diagram w E D [ a ,b] there exists a unique transition measure p = ,u(W). It is supported by the interval [a,b]. The map w H p(W) establishes a homeomorphism of the space V [ a ,b] onto M [ a ,b]. Let us preface the proof with several examples.
$2. CONTINUOUS DIAGRAMS A N D
R-FUNCTIONS
153
EXAMPLE1. The distribution dp(u) = (2n-)-ld=du with the semicircle density on the interval 1-2,2] is the transition measure of the diagram R (Example 1, Section 2.2), since for u 2
EXAMPLE 2. The transition measure of the "triangular" diagram from Example 2, Section 2.2 is the beta distribution with density &(u)
sin n-p n-
= -(1
+ u)-q(l
-
on the interval [-I, 11. Indeed, from the basic integral representation of the hypergeometric function and the identity
we have
EXAMPLE 3. Comparing (2.2.4) and (4.2.9), we conclude that the transition measure p of a rectangular diagram w is discrete, and the defining relation (4.2.10) takes the form
The weights pk of the measure p are located a t the minima xk of the diagram w and coincide with the coefficients of the partial fraction expansion. The left-hand and right-hand sides of (4.2.11) uniquely determine each other; the positivity of the coefficients pk is equivalent to the separation property of the extrema (4.2.3). Let us write the transition formulas in an explicit form:
and n-l
n
Let us show that the bijection established by these formulas between the subsets Do and M o can be extended by continuity to a homeomorphism of D[a, b] onto M [a, b]. For this purpose it is convenient to use moments.
154
4.
YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS
2.4. Moments of a continuous diagram. A finite measure p on the interval [a, b] is uniquely determined by its moment sequence {h,}r==,, where
(this fact is known as the uniqueness of solution of the Hausdorff power moment problem). The weak convergence of measures p, -+ p is equivalent to the convergence of moments: hk(pn) -+ hk(p) for all k E N. Let Rm be the space of sequences with the componentwise convergence topology. Then (4.2.14) defines a homeomorphism of the space M [ a , b] onto the closed subset M C RM of moment sequences. The sequences from M can be characterized by a series of algebraic inequalities b which express the nonnegativity of the integrals Ja(b - t)"t - a)" dp(t) in terms of moments; however, we will not need this characterization. The moment generating function of a measure p is the series
which converges for sufficiently large lul. The generating function of p and its R-function
are related by the formula H ( u ) = u R(u); that is,
Let us now describe the moment embedding of the space of diagrams V into
Rm. By definition, the moments of a diagram w E V[a, b] are the numbers {pn)r?'_,, where
Let us call the series
the generating function of moments of the diagram w. This series converges for sufficiently large lull and S(u) =
;J.
d(w(t) - It0 = t-u
lb
t-u
The map w ct { p , ( ~ ) ) ? = ~defines a homeomorphism of the space V[a, b] onto a closed subset 2) c RM. Indeed, since the total variation of the charge Ida( = w(0) is uniformly bounded for w E D[a, b], the convergence of moments is equivalent to the weak convergence of charges. In view of the uniform Lipschitz condition (i) from the definition of a diagram, this implies the uniform convergence of diagrams on the interval [a, b] (and hence on the whole line).
52. CONTINUOUS DIAGRAMS AND R-FUNCTIONS
The relation m
00
(4.2.20)
H(u) =
hn uPn = exp n=O
E u - " = exp S ( u ) n=l n
defines a homeomorphism Exp: BM + WM which maps a sequence {p,) t o {h,) ({h,) and {p,) can be expressed polynomially in terms of each other). It follows 0 )and the map Exp sends from (4.2.16), (4.2.19), and (4.2.11) that ~ ~ ~ (=3Mo, the moments of a diagram w to the moments of the distribution p = p ( W ) . Since the set vo of moment sequences of rectangular diagrams is dense in V , and the corresponding set Mo for finitely supported measures is dense in M , we conclude that E ~ ~ ( = v )M , and the theorem of Section 2.3 follows. Denote the constructed homeomorphism by EX^,,^: D[a, b] + M [a, b] . The centre and area of a diagram are expressed in terms of its moments as follows:
For z this follows from (4.1.18), and for A, by integration by parts:
2.5. Behaviour of diagrams under translations and homotheties. If a rectangular diagram w is constructed from a pair of sequences (1.1.2), then the function
with c > 0 is also a rectangular diagram; it corresponds to the sequences
The centre of the new diagram is translated t o the point 5 = (z - b)/c, the area is reduced by c2, i.e., A = A/c2, and the charge and R-function are given by
All these formulas (of course, except (4.2.23)) remain valid for arbitrary (continuous) diagrams. 2.6. Computations for the arcsine law and semicircle distribution. Among all diagrams, the most important for us is the diagram
This diagram is centred at the origin and has unit area.
$3. T H E KREIN CORRESPONDENCE
157
PROOF. Clearly, the odd moments vanish. The substitution u = 2 sin cp yields
From the binomial identity we have
Substituting s = 212 and using formula (4.2.28), proved above, we obtain
COROLLARY. The transition measure of the diagram R is the semicircle law (4.2.27):
& (z)
often arise in combinatorial problems, Note that the numbers cm = and are known as the Catalan numbers. In some cases it is more convenient t o use the R-function
instead of the generating function (4.2.15). The R-function of the diagram R is
53. The Krein correspondence In this section we describe the bijection between continuous diagrams and probability measures which arises as the continuous extension of the correspondence between Young diagrams and their transition distributions in the Plancherel growth process. Then we give alternative descriptions of the same correspondence discovered by various authors in completely different problems. One of the best-known descriptions is based on M. G. Krein's ideas, so we call our bijection the Krein
correspondence. The relation of the diagram approach t o other descriptions is useful in both directions. It allows us to generalize combinatorial results concerning permutations, and on the other hand, to use the properties of the Krein correspondence in asymptotic problems of the representation theory of symmetric groups. Section 3.1 contains an axiomatic description of the Krein correspondence. In Sections 3.2 and 3.3 we describe the Hausdorff and Markov moment problems, and the original construction from [47] relating them. In 52 above we obtained our bijection by passing to the limit from the Plancherel measure of the symmetric group. In Section 3.4 this bijection arises in a statistical problem concerning the
158
4.
YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
distributions of integrals over random Dirichlet measures. In 3.5 we mention some explicit formulas for the Krein correspondence. In Section 3.6 we show that the transition probabilities of the Plancherel measure of 6, tend, with the growth of the Young diagram, to the semicircle distribution. Here we make substantial use of the theorem on the limit shape of large random diagrams obtained in [12, 1381. 3.1. An axiomatic description of the correspondence. Denote by M = M [ a , b] the space of probability measures on the interval [a,b] endowed with the weak topology, and let Mo c M be the dense subset of finitely supported measures.
DEFINITION. The R-function of a measure p E M [ a , b] is the function Rp(u) =
lb
u-t'
Like the R-function of a diagram, it is defined and holomorphic outside the interval [a, bl.
EXAMPLE. The R-function of a discrete measure p E Ma with atoms of weights pk at points xk, k = 1 , . . . , n, is
Recall that the R-function of a diagram w E V is the generating function of its moments,
'J
Rw(u)= - exp u 2
d(w(t) - Itl) t-u .
The R-function of a rectangular diagram w E Do can be written in the form
where x l < yl < . . . < y,-1 < x, are the extrema of the diagram. According to the theorem of Section 2.3, the bijective correspondence between diagrams w E 2) and their transition measures p = pW E M can be defined axiomatically by the identity
which is valid for the values of u outside the base interval [a, b]. For rectangular diagrams the identity reduces to the formula
Thus (4.3.5) is a generalization of the partial fraction expansion.
53. T H E KREIN CORRESPONDENCE
159
3.2. The A. A. Markov moment problem. The Hausdorff moment problem consists in describing the set of moment sequences
where p is an arbitrary Bore1 measure on the interval (and in recovering the measure from its moments). The solution of this problem is well-known. The less-known Markov moment problem consists in describing the set of moment sequences
where the measure dr(t) = r l ( t ) dt is absolutely continuous and has a bounded density 0 5 r l ( t ) 1. We are mostly interested in the relation between the Hausdorff and Markov moment problems: it turns out that the latter can be reduced in a nontrivial way to the former [47]. To this end, let us introduce the moment generating function
<
M
-n
t-u
n=l
and observe that in view of our assunlptions -T < Im S(u) < 0. Hence the exponential H(u) = exp S(u) is a resolvent function; that is to say, it is holomorphic in the upper half-plane and Im H ( u ) 0 for I m u > 0. It follows from a known integral representation of such functions that there is a unique probability measure p such that
<
The moments from the Markov problem can be easily expressed in terms of the moments of this measure p: it suffices t o rewrite the defining relation (4.3.11) in the form
Thus the description of Markov moment sequences is reduced to the well-studied Hausdorff problem. The relation between the measures d r and dp becomes especially simple in the case when the density r l ( t ) is equal to one on the subintervals (0, x l ) , . . . , (ynPl,x,) and vanishes on the complementary subintervals (XI,yl), . . . , (x,, 1). Then H ( u ) is a rational function, and the measure p attaches the weights
to the points xk, k = 1 , . . . ,n. The defining identity (4.3.11) determines the partial fraction expansion of H (u)/u:
53. THE KREIN CORRESPONDENCE
161
For a rectangular diagram w E Do the Rayleigh measure attaches unit weights to the points of minima xk and negative unit weights -1 to the points of maxima yk. The moments of a diagram w coincide with the moments of its Rayleigh measure C.
The defining identity (4.3.5) takes the form
The characteristic property of Rayleigh measures is given by formula (3.2) from $4 of the Introduction. The third approach to the Krein correspondence is related to the Dirichlet random measures arising in nonparametric statistics (see [91]). As a simple example, consider a discrete probability distribution T with atoms of weights T I , . . . , T , at real points X I , .. . , x,. We associate with this distribution the probability measure on the (n - 1)-dimensional simplex
with the Dirichlet density
Denote by p the distribution of a linear functional
on A,. It is known (see [80, 821) that the original distribution T and the distribution p are related exactly by the Krein formula (4.3.17). In a completely analogous way, we may associate with each probability measure r on a real interval a random Dirichlet process whose realizations are random probability measures (see [131],Chapter 9). The expected value (4.3.18)
X=
/
xda(x)
with respect to the random measure a is a random variable; denote its distribution by p. Then T and p are related by (4.3.17). It is known (see [82])that the random variable X is finite with probability one if and onlv if
Another general fact is that the distribution of X is absolutely continuous provided that T is a positive measure with nonzero variance. Let us give a number of examples, reproduced from [82].
EXAMPLE 1 . Let T be the measure with atoms of weights 1 - 0,O at the points 0 , l . Then the random variable (4.3.18) has the beta distribution P(1 - 0,O). In this example the diagram w is triangular, cf. Example 2 from Section 2.2.
4.
162
YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS
EXAMPLE 2. Assume that T is the uniform distribution on the interval [O,l]. Then the distribution of X has the density e - (1 - x ) - ' + ~ xPx sinn-x, 0 < x < 1. n-
EXAMPLE 3. Let T be the distribution function of eC / (1+eC), where C obeys the Cauchy distribution. Then the distribution of (4.3.18) is uniform on [0, 11. EXAMPLE 4. If T is the Cauchy distribution, then so is p. This property uniquely characterizes the Cauchy distribution.
EXAMPLE 5 . Assume that T is a discrete probability measure with atoms of ~k a t points xk, k = 1 , . . . ,n . Then the transition distribution p has the
weights density
on any interval [a, b] containing the support of T. This example generalizes Example 1 above and Example 2 from Section 2.2; in this case the diagram w is piecewise linear and convex. 3.5. Explicit formulas. The transition probability p = p(W)for general diagrams w E D is defined implicitly by the identity (4.3.3). We would like to have explicit formulas similar to (4.2.12) in the case of rectangular diagrams. Approximating a diagram w E 2) by rectangular diagrams and passing to the limit in (4.2.12), we easily obtain the following expression for the weights of the discrete part of the transition measure:
a(w){t}= exp
Jr:
b
-,dot (u) u-t
where a t ( u ) = (w(u) - IU - t1)/2. Let us also give a hypothetical formula for the density cp(W) of the absolutely continuous component of the transition measure dp(w)(t)= cp(W)(t) dt of w: cp(W)
;b
dw(u)
1 n-wl(t) exp .fa (t) = - cos n2 J(b-t)(t-a)
3.6. The semicircle law for the transition probabilities of the Planchere1 growth process. I11 52 of Chapter 3 we showed that in the course of the random Plancherel growth almost all Young diagrams approach (after an appropriate scaling) a common universal shape close to the diagram R. More precisely, the following theorem holds.
THEOREM ([12], [138]). For almost all (with respect to the Plancherel measure M) random tableaux t = (A1,. . . , A N , .. . ) E 7 the limit 1 lim -AN ( u n ) = R(u) N + a
% ,
exists uniformly in u E R. Using the continuity of the correspondence between diagrams and transition probabilities, we can restate this result as the "semicircle law" for the transition probabilities of the Plancherel measure.
54. INTERVAL SHRINKAGE PROCESS
163
COROLLARY. Let p ~ be the transition distribution corresponding to a Young diagram AN. Then for almost all (with respect to the Plancherel measure) random tableaux t = (A1, . . . , AN,.. . ) E 7 the limit lim p N { r : r
3 0 ' N
exists for all lul
=
-
< 2.
The proof follows immediately from the previous theorem, the continuity of the Krein correspondence, and the computations of Section 2.6. 54. Interval shrinkage process Here we describe a stochastic algorithm of shrinking the interval I = [a,b] associated with a diagram w E D [ a ,b] defined on this interval. This algorithm is a Markov process of transition from a subinterval [a, P] C I to a smaller subinterval; a realization of the process is a sequence {[a,,P,]),",, of nested intervals. With probability one the intersection (-)[a,,P,] consists of a single point X. The main result says that the distribution of the random variable X coincides with the transition measure p(W)= Exp(w) defined in 52. We start with the simplest situation, when w is a rectangular diagram. 4.1. The discrete case. Consider a pair of interlacing sequences (4.4.1)
x1 < y l <
22
< . . . < x,-1 < y,-1 < x,,
and let p be the measure with atoms of (positive) weights
a t the points xk, k = 1 , . . . , n. These numbers are the coefficients of the partial fraction decomposition
Multiplying both sides by u and letting u -+ m, we see that C pk = 1. There is an elegant Markov chain, the interval shrinkage process, whose final probabilities coincide with (4.4.2). The state space of this chain is finite and consists of all intervals of the form [xi,xj], where 1 5 i 5 j 5 n. The initial state is the largest interval [xl, x,] . Assume that the current state is an interval [xi,xj]. In order to obtain the next state, pick a point y uniformly distributed on [xi,xj]. It falls into one of the subintervals into which the points (4.4.1) divide the interval [xi,xj]. There are two cases:
for some k . By definition, the next state is the interval [xi,xk] in the first case, and [xk, xj], in the second case. At each step of the process the endpoints of the interval approach each other; after at most n steps the interval degenerates into a point, and the process stops.
YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS
4.
164
PROPOSITION. The probability that the interval shrinkage process stops at a state [xk,xk] is equal to pk.
<
Given a triple i k 5 j, let p?) be the probability that the process stops at [xk,xk] provided that it started from [xi,xj], and denote by R?) and the probabilities of reaching [xk,xk]starting from [yiPl,xj] and [xi,yj], respectively. By the formula of total probability we have PROOF.
where zij yields
=
xizixs
LY)
ys. Again applying the formula of total probability
-
R?) = xi
- Yi-1
zij - Yi-1
(ij)
pk
+
zij - xi Rf+l,j) 1 zij - Yi-1
These formulas express the numbers p p ) , R?), L?) in terms of the same quantities with smaller values of j - i . Starting from the obvious values p F k ) = R F k ) = L f k ) = 1, we obtain by induction on j - i that
R?) = xk
- Yi-1
zij - Yt-1
For i = 1 and j
=n
(ij) =
pk
xk - xi-1
pt-l,j)
Zi-1,j - Xi-1
7
the first formula yields (4.4.2), as required.
The interval shrinkage algorithm is a version of the hook walk algorithm which is well-known in the combinatorics of permutations; a q-analogue of this algorithm was constructed in [36], see also $7 of Chapter 3. 4.2. The interval shrinkage process in the general case. The above procedure allows us to construct, given a rectangular diagram (4.4.1), a random variable distributed according to its transition measure. In the case of a general diagram w E D[a,b] the construction is analogous, but the state space of the process is continual and the random variable X with distribution p(W) arises in the limit of infinitely many iterations. Let us proceed to a precise description of the general interval shrinkage process. By definition, the states of the process are all closed subintervals [a,P] C I of the base int,erval I = [a, b]. The initial state is the interval [a,b]. Each step of the process consists of two independent random procedures: 1) cutting the current interval [a,p] by a random point y; and 2) choosing one of the resulting parts [a,y], [Y, PI. The distribution of y is always the same: the uniform distribution on the corresponding interval [a,PI. The controlling diagram w E D[a, b] determines only
$4.
INTERVAL SHRINKAGE PROCESS
165
the probabilities q, p of choosing, respectively, the left or the right of the subintervals [a,y], [y,p]. By definition, these probabilities are equal t o
Iterating this procedure yields a countable sequence of nested intervals (4.4.6)
[a, b] 3 [al,Pl]3 . . . 3 [a,, p,] 3 . . . ,
their endpoints tending with probability one to a common limit (4.4.7)
X
= lima, = limp,.
THEOREM.Let X be the random variable (4.4.7) determined by the interval shrinkage process. If this process is controlled (according to (4.4.5)) by a diagram w, then the distribution of X coincides with the transition measure p ( W ) of this diagram. In the special case of the triangular diagram wp from Example 2 of Section 2.2, the probabilities (4.4.5) are constant and do not depend on the current state [a,PI. The limiting variable X has the beta distribution with density sin ~p f (x) = -(1 n-
+ x ) - ~(1- x)-P.
The simplest case p = q = 112 is worth stating separately. COROLLARY. Consider the random sequence of intervals
defined by the following rule. The interval [an+l, pn+l]is obtained from [a,, p,] by dividing it into two parts by a uniformly distributed point, and independently picking one of these parts with equal probabilities. Then the limit X = lima, = limp, exists almost surely, and its distribution has the inverse semicircle density
on the interval [- 1,1]. Let us give an alternative description of the interval shrinkage process which makes it closer to the hook walk algorithm from [36]. From each point 0 lying above the graph of a diagram v = w(u), draw the lines with slopes f1; let them meet the graph at points M,, Mp with abscissas a < /3. The union H,,p of the intervals M,O (leg) and OMp (arm) is called the hook at the point 0 . The hook is uniquely determined by the interval [a, /3] (Figure 23). The hook walk process starts from the hook Ha,bover the initial interval [a,b]. We pick a random point uniformly distributed on Ha,b and take the hook associated with this point; then we pick independently a random point on the hook Ha1,p, and take the corresponding hook H,,,p,, and so on. The hook walk is equivalent t o the interval shrinkage process, as can be seen from the following observation. Let M be a point on the graph v = w(u) with abscissa u E [a,/3]. Denote by B the point of intersection of the arm of the hook with the line V =
4.
YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
FIGURE 23. The hook walk algorithm and the interval shrinkage process.
+
w(u) (U - u), and by A the point of intersection of the leg of the same hook with the line V = w(u) - (U - u) (Figure 23). Let
+
+
Then the abscissas of the points A and B are r ( a ) 1(u) and r(u) 1(P), respectively. If A (respectively, B) is uniformly distributed on the leg (arm) of the hook H,,p, then u is distributed with density lt(u) (respectively, rt(u)). Since lt(u) = (1 w1(u))/2 = q and rt(u) = (1 - wt(u))/2 = p, the equivalence of the shrinkage process and hook walk is obvious. Informally speaking, it is convenient to think of dr(u) as the measure of illumination of the diagram from the northeast, and of dl(u), as the measure of illumination from the northwest. These two measures already appeared in our considerations. Indeed, T = lt(u) is the distribution function of the Rayleigh measure, and the density f (u) = r t ( u ) appears in the (0, 1)-Markov moment problem, see [471.
+
4.3. Proof of the theorem. Denote by p,,p the resulting distribution of the interval shrinkage process with initial interval [a,PI. Let R (respectively, L) be the conditional distribution of the random variable (4.4.7) provided that at the first step we have chosen the right (respectively, left) subinterval of [a,b]. Averaging over p,,p, we can write the distributions L, R as
The key property of the distribution p = pa,b, which reflects the recursive character of the interval shrinkage process, is expressed by the identity p
b-z = L
z-a ~ + R.
~
54. INTERVAL SHRINKAGE PROCESS
167
In order to prove the theorem of Section 4.2, we will check that (4.4.11) is also satisfied by the transition probabilities palo corresponding to the restrictions wale of w t o subintervals \a,P] c I. So let pa)o = E X ~ , , ~ ( W ~be? P the ) transition measure of such a restriction, and consider, instead of (4.4.9), the integral
LEMMA1. The measure L is absolutely continuous with respect to p, and the corresponding density is linear: dL(u) - b - u dp(u) b-z'
PROOF.Consider the function
445) =
lx
t-x
dpa7l ( u ) = enp
lxs.
Its differential equals
dmt ( x ) =
x-t
dl(x)=
(l
)
d,~~,~(u) dl(x)
u-t
By the meaning of the integral (4.4.12), the Stieltjes transform of the measure ( b - z ) d L ( u ) equals
(b-z)
Jb%=S,"(l a
t-u
d ~ ~ , ~ ( u ) t-u )dl(%)=-
dbt(s)=l-$t(b).
It coincides with the Stieltjes transform of the measure ( b - u ) d p ( u ) :
and the lemma follows from the inversion theorem for Stieltjes integrals.
LEMMA2. Define a measure R as a n integral of the distributions ,u"$~:
Then R is absolutely continuous with respect to p density equals dR(u) u - a dp(u) z-a'
=
palb, and the corresponding
The proof of this lemma reproduces the previous argudent, with the function -2 & ( x ) replaced by
168
4.
Y O U N G D I A G R A M S IN P R O B L E M S O F A N A L Y S I S
We see from Lemmas 1 and 2 that the transition measures pQ>Pdefined by the same diagram w on different subintervals are related by the integral formula
which is equivalent to the defining equation (4.4.11). The theorem follows.
55. Differential model of growth of Young diagrams 5.1. The idea of a differential model. The starting point for this section is the result on the asymptotics of random growth of Young diagrams obtained in [12, 1381. Those papers dealt with a discrete time Markov process whose state space is the set of all Young diagrams. At each step of the process a new random square is attached to the current diagram, and the definition of transition probabilities is motivated by asymptotic problems of the representation theory of symmetric groups, see [17]. Keeping in mind the representation-theoretic origin of this process, we call it the Plancherel growth process. As a diagram grows, its area increases to infinity. If we uniformly scale the diagram so that the rescaled diagram has unit area, the edge of the diagram looks more and more like a continuous curve. The key point of the theorem discovered and proved in [12, 1381 is the following. In the course of the Plancherel growth process almost all Young diagrams (after the normalization of the area) become uniformly close to a common universal curve. In a natural coordinate system the equation of this curve is
Figure 8 on p. 35 shows the graph of this curve, along with a random diagram with a hundred squares obtained by a computer simulation of the Plancherel growth process. For fifteen years the appearance of the curve (4.5.1) seemed to be a specific isolated result. At present the situation is different: it turned out that the same curve arises naturally in various problems of one-dimensional mathematical physics and function theory (see [118], [38], [37]). As shown in [38],the curve (4.5.1) describes the common asymptotics of separation of roots for a wide class of orthogonal polynomials (including the classical polynomials of Chebyshev, Hermite, and Laguerre). The same curve describes the typical character of mutual separation of frequencies under a random linear constraint in a wide class of linear mechanical systems (see [38]). Below we establish new characterizations of the curve (4.5.1) which relate two contexts where it appears: the Plancherel growth of Young diagrams and one-dimensional problems of mathematical physics. The main idea is to replace the original discrete time random process by a continuous time deterministic process with the same asymptotic behaviour as t + OC).
Roughly speaking, the elimination of randomness is achieved as follows. Given a certain state of the Plancherel growth process, assume that the new square can be attached to the current Young diagram a t points X I , .. . ,x,, with probabilities p1,. . . , p,. Then, instead of randomly attaching the entire square to one of these
55. DIFFERENTIAL MODEL O F GROWTH O F YOUNG DIAGRAMS
169
points, we would like to attach simultaneously a part of the square proportional to the probability pk t o each point xk. In order to implement this idea we substantially extend the state space by considering along with Young diagrams the limiting curves of the form (4.5.1). See $2 for a precise definition of continuous diagrams. The extended space of diagrams V is an infinite-dimensional compact space which is canonically homeomorphic to the space of probability measures on an interval. The correspondence which associates with a diagram w E V a probability distribution pw is by no means trivial. It was implicitly used in connection with the Markov moment problem [47] (cf. [37]). However, in our context the map w H p, arises quite naturally: if w E Y is a Young diagram (maybe rescaled), then p, is its transition distribution in the Plancherel growth process. The correspondence w H pw can be uniquely extended by continuity to general diagrams w E V (see [I181and $3). Another interpretation of the correspondence w H pw is related to the partial fraction expansion ($2). The history of growth of a general diagram (an analogue of a Young tableau) is described by a curve w(.,t), 1 < t < oo, in the space 'D. The diagrams w(.,t) are assumed to increase (with respect t o the inclusion of subgraphs) with t. The infinitely thin layer between the graphs of the diagrams w(u, to) and w(u, to dt) determines a one-dimensional distribution with density CW:(U,to) (where C is a normalization constant); it is natural t o think of this distribution as a tangent vector to the tableau w(., t) a t the point t = to. Equating the tangent and transition distributions, we arrive at the basic dynamic equation
+
A more closed form of (4.5.2) is given in (4.5.10) below. Equation (4.5.2) means that the diagram w(., t) always grows in the direction of its transition measure. Thus we can expect that the asymptotic behaviour of solutions of (4.5.2) is related to that of the Plancherel growth. Our main result (the theorem of Section 5.6) completely justifies these expectations: the diagram R from (4.5.1) is the unique, up to scaling, fixed point of (4.5.2), and all other solutions are asymptotically attracted by this point. In the course of our analysis we rewrite (4.5.2) in several equivalent forms. Surprisingly, one of the equivalent restatements of (4.5.2) is the quasilinear firstorder equation
which describes the free motion of a one-dimensional medium of noninteracting particles. This is one of the simplest equations illustrating nonlinear phenomena, in particular, the appearance of shock waves. In terms of equation (4.5.3), the curve (4.5.1) corresponds to the automodel solution
where (4.5.4) It follows from the main theorem that if the initial wave R(x, 1) is analytic at infinity, then, in the course of its evolution controlled by equation (4.5.3), its shape
170
4.
YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS
in a neighbourhood of the point x = cc tends, as t + ca,to the shape of the automodel solution (4.5.4). The section is organized as follows. In Section 5.2 we define a continuous analogue of Young tableaux, and in Section 5.3 we introduce the differential equation which controls the growth of continuous diagrams. Then we prove three theorems characterizing the diagram R. I11 Section 5.4 we show that this diagram determines the unique automodel solution of the basic equation. According to Section 5.6, all other solutions are asymptotically attracted by this solution. In Section 5.5 we claim that R is the unique (up to scaling) diagram whose transition distribution coincides with the radial distribution (defined in Section 5.5 from simple geometric considerations). 5.2. Continuous tableaux. Like Young diagrams, general continuous diagrams can be ordered by inclusion of subgraphs: wl + w2 if D,, c D,,. A tableau is a continuous D-valued map defined on some interval [to,m) and strictly increasing on this interval. Thus a tableau is a family of diagrams w(.,t) increasing in t. All these diagrams have a common centre z(t) = z ; without loss of generality we assume that z = 0. The area A(t) continuously increases with t ; it will be convenient to take it as a parameter, assuming that A(t) = t. Given a tableau w(u, t ) , denote by u(u, t) = (w(u, t) - 1u1)/2 the corresponding charges (see Section 2.2). Taking into account our assumptions on the parameter t and the normalization of the area, we conclude that
DEFINITION. The function T ( u , t ) = u:(u,t) is called the tangent density of - at the point w(., t). the tableau {w(.,t))t>to Differentiating (4.5.5) in t, we obtain that T(u, t) is indeed the density of a probability distribution, which we call the tangent distribution: (4.5.6)
J
T(u, t ) du = At(t) = 1.
Let us compute the moments of the tangent distribution. LEMMA.Let pn(t) be the moments of a diagram w(u, t ) in a tableau {w(., t))t>to. Then the moments of the tangent density can be written in the form
PROOF.By the definition of the tangent density,
and the lemma follows from the equations
55. DIFFERENTIAL MODEL O F GROWTH O F YOUNG DIAGRAMS
171
5.3. The basic dynamic equation. Consider a growing diagram, that is, a tableau {w(.,t)It2t,, and let pt = P,(.,~)be the transition measure of the diagram w(u, t). We require that the diagram w(u, t ) grow in the direction of its transition distribution:
Relation (4.5.8) should be regarded as a n evolutionary equation in the infinitedimensional phase space of diagrams V ,where the correspondence w H p, determines the canonical vector field (more precisely, the field is defined on the subset of diagrams with absolutely continuous transition measures). Using the definition of the transition measure, we can rewrite (4.5.8) in a closed form: cr;(u, t) du = exp for sufficiently large x. 1 - u/x Let us give two other forms of the basic dynamic equation (4.5.8): in terms of the moments pn = pn(w) of the diagram and in terms of the generating function
J
(for a fixed t this function coincides with the R-function from 53, formula (4.3.3)). PROPOSITION. The following forms of the basic dynamic equation are equivalent:
PROOF. The equivalence of (4.5.10) and (4.5.11) follows from the lemma of Section 5.2. Consider the series
Since pl (t) = z = 0, (4.3.5) implies S ( x ,t ) = log(xR(x,t)), whence Si(x, t) = R-'(x, t ) R:(x, t). Comparing the series
we see that (4.5.11) and (4.5.12) are equivalent too. Note that (4.5.12) is one of the simplest examples of quasilinear first order partial differential equations; among other applications, it is used as an illustration of
4.
172
YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
nonlinear phenomena such as breaking of wave crests or the birth of shock waves. In one of its numerous interpretations, (4.5.12) is regarded as the equation describing the free motion of a medium of noninteracting particles. In this case R(x, t ) is thought of as the velocity of the particle located a t point x at time t.
5.4. A u t o m o d e l solutions. For every diagram w of unit area A(w) = 1, the formula
determines a tableau. Tableaux of this form will be called automodel. Let us show that the unique automodel solution of the basic dynamic equation is generated by the diagram R. THEOREM.Let w(u) be an arbitrary diagram of unit area A(w) = 1, and w(u,t) = \/tw(u/\h), the corresponding automodel tableau. If its charge o ( u , t ) satisjies (4.5.10), then w(u) = R(u).
PROOF.Consider (4.5.10) in the equivalent form (4.5.12). It is easy to check that the moments p,(t) of the diagram W(U,t) can be expressed in terms of the moments p, of the initial diagram w(u) in the form
By the homogeneity of formulas (4.1.17), we have a similar relatiorl for the moments of the transition measures: h,(t) = tn/2h,. Thus the function (4.5.9) is equal to R(x, t ) = R(x/&)/&. Substituting this expression in (4.5.12) yields, after a simple calculation, the relatiorl (xR(x))' = (R2(2))' ; hence xR(x) and R2(x) differ by a constant. Comparing the constant terms, we corlclude that R2(x)- xR(x) 1 = 0. The only solutiorl of this equation vanishing as x + +co coincides with the generating function of the diagram R. The theorem follows.
+
Irlformally speaking, we have shown that R(u) is the only diagram that remains self-similar (i.e., does not change its shape up to rescaling) in the course of the growth in the direction of its ow11 transition measure. For automodel tableaux we have
The expressiorl in the right-hand side is determined by the diagram w(u) = w(u, t) for a fixed value o f t ; it has a simple geometric interpretation which will be discussed in the next section.
5.5. T h e radial d i s t r i b u t i o n of a diagram. Given a pair of arguments a < p, denote by Ma, Mp the corresponding points of the graph of a diagram w, and let A,P be the area of the curvilinear sector in between the rays OM,, OMo (where 0 is the origin), and the arc M,Mo of the graph (see Figure 24). 1
A""
P (W (u) -
w l ( u ) ) da =
(o(u) - uol(u)) du.
55. DIFFERENTIAL MODEL O F GROWTH O F YOUNG DIAGRAMS
173
FIGURE24. Radial distribution of a diagram. PROOF. Consider the ray exiting the origin 0 a t an angle 0 and meeting the graph of the diagram at a point M. Denote by r ( 0 ) the length of the segment O M , and by u the abscissa of M ; u and 0 uniquely determine each other: 0 = B ( u ) . Integrating in polar coordinates yields Aa8 = r 2 ( 0 )dB. Substituting 0 = 0 ( u ) gives ctg B = u / w ( u ) , whence
-a
[~(u )u w l ( u ) ]d u
= w 2 ( u )d ( u / w ( u ) )=
~,e( $
- w 2 ( u ) sin-2 0 d0 = - r 2 ( 0 ) dB.
The second claim of the lemma follows immediately from the definition of the charge of a diagram. Let us define the radial density of a diagram w E
D by the formula
It follows from the lemma that
so that the radial density determines a probability distribution. We call it the radial distribution.
PROPOSITION. Let pn be the m o m e n t s of a diagram w E D. T h e n the m o m e n t s of its radial distribution are equal t o
PROOF. We have
As a moment generating function for the radial distribution we may take (assuming pl = z = 0 ) the function
174
4.
YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
THEOREM. Assume that the transition distribution of a diagram w coincides with its radial distribution, dp,(u) = p,(u) du. Then w coincides, up to similarity, with the diagram R for all u E R:
where A = A(w).
PROOF.Let us express the equality of moments of the distributions under con= h,, in terms of the function R(x) = hn x-("+l). sideration, ~ , + ~ / 2 ( n + l ) A By (4.5.17) we have
En,,
(since xR(x) = exp S(x)). Comparing with the expansion
+
yields (xR)'/R = 2AR1, or (xR)' = (AR2)'. Thus AR2 - XR C = 0 for an appropriate constant C. Comparing the lowest terms of the expansions xR(x) = 1
+ t/x2 + . . .
and
AR'(X)
= A/x2
+ . .. ,
we find that C = 1, so that
for x Finally, R(x) = (x - d=)/2A remains to observe that the function
> 2 a (since R(x) + 0 as x + + m ) . It
coincides with the analogous generating function in the right-hand side of (4.5.18). Note that according to (4.5.14) the tangent distributions of an automodel tableau coincide, at each diagram, with its radial distribution. It is easy to check that this condition characterizes the class of automodel tableaux. This fact can be regarded as "Kepler's second law for the transverse growth of trees". Indeed, assume that the graph of a diagram w describes the profile of a cut of a tree trunk. The shape of this profile will be stable up to similarity if and only if the intensity of appearance of new cells within an arc is proportional to the area of the sector supported by this arc. 5.6. The asymptotics of the general solution. We have shown in Section 5.4 that the diagram R is, up to similarity, the only fixed point of the basic dynamic equation (4.5.8). Let us now check that this solution is stable (attracting all other solutions). More precisely, each solution w(u, t) of (4.5.8) (with arbitrary initial conditions), after the normalization of its area, converges uniformly to R(u) ast+m.
55. DIFFERENTIAL MODEL O F GROWTH O F YOUNG DIAGRAMS
175
THEOREM.Assume that the charge a ( u , t) of a tableau w(u, t) satisfies (4.5.10). Then
unzformly in u E R. PROOF. The moments 5, (t) of the normalized diagram G(u, t ) = w(u&, t ) / & are easy to express in terms of the moments of the diagram w(u, t):
p, ( t )= t Let us find their asymptotics as t
-+
(t).
m , using (4.5.11).
LEMMA.If the functions p,(t), n
=
1 , 2 , .. . , satisfy
then there exist constants {P,),",~ such that
PROOF. The first equations of the system (4.5.20) are p;/3 = 0, pk/4 = 3(p? + ~2112,
+ pk/6 = 5 ( ~ + ? 6 ~ ?+ ~ 32 ~ + ; 8 ~ 1+ ~ 63~ 4 ) / 2 4 . Pk/5 = 4(P? + 3 ~ 1 ~ 22~ 3 ) / 6 , Successively solving them, we find that
where the c k are arbitrary constants. Since the right-hand side of (4.5.20) is homogeneous, the lemma follows easily by induction. To complete the proof of the theorem, it remains to check that the limiting values p, in (4.5.21) coincide with the moments of the diagram a. This can be done most conveniently by going over to equation (4.5.12). h, ( t ) ~ - ( ~ +is' )constructed from the normalized If the function R(X,t) = diagram G(u, t) = w(u&, t)/&, then
zr=o
176
4.
Y O U N G DIAGRAMS IN P R O B L E M S O F ANALYSIS
Substituting this expression into (4.5.12) yields, after simple calculations, the equat ion
Let us write the function R in the form
+
(4.5.23)
~ ( xt), = ~ ( x ) ~ ( xt ), ,
where ~ ( z =) C,"==, h , ~ - ( ~ + ' ) the , coefficients h, being related to pn in the usual way. Then, by the lemma, lim ~ ( xt), = 0,
t--*a
lim t ~ k ( xt), = 0.
t-m
Substituting (4.5.23) into (4.5.22) and letting t -+ m , we see that ZR - R2 = const, and the proof can be completed like those of the theorems of Sections 5.4 and 5.5 by using the quadratic equation R2(x) - z ~ ( z ) 1 = 0 and the formula for the generating function of the transition measure of R.
+
$6. Plancherel growth and semicircle diffusion
The transition distribution of the arcsine law R given by (4.5.1) is the semicircle distribution with density J-12~. In this book the arcsine law has appeared repeatedly in various contexts. The semicircle law also appears in many situations, of which we will consider only two: Wigner's semicircle law for the spectra of random matrices, and Voiculescu's free probability theory. In this section we consider the relation between the arcsine law and the semicircle law in more detail. 6.1. Wigner's semicircle law. Another context where the curve R appears is related to the spectral theory of random matrices and Wigner's semicircle law. Let A(") be a symmetric matrix of order n with characteristic polynomial
Denote by p(,) the discrete distribution with atoms of equal weights p p ) = l / n at the characteristic roots xk, k = 1,.. . , n. Denote by
the distribution function of the measure p("), and by F ( x ) the distribution function of the measure dp(u) = ( 2 ~ ) ~ ' du on the interval [-2,2]. We are interested in the case when the matrix A(,) is random; in this case the polynomial P,, its roots, the measure p(,), and the function F, are also random.
THEOREM (see [SS]). Let the matrix entries a/;) for i = 0 and ~(a!;)) = 1. Then, for every r > 0,
<j
be i.i.d. with ~(a:;))
56. PLANCHEREL GROWTH AND SEMICIRCLE DIFFUSION
This result can be restated in terms of diagrams. As above, let X I , . . . , x, denote the characteristic roots of the matrix ~ ( ~ and 1 , let y l , . . . , y,-1 be the extrema of the characteristic polynomial Pn(x) of A("); these two sequences interlace. Let 3, be the rectangle diagram constructed from the pair of sequences 2 1 < y1 < . . . < yn-1 < x,.
PROPOSITION.Under the assumptions of the theorem,
a s n - + m for every6 > 0 . The equivalence of this result and Wigner's theorem is a consequence of the remarkable Krein correspondence between diagrams and probability measures. Indeed, since
the diagram an constructed from the zeros and extrema of the polynomial corresponds to the uniform distribution on its roots, and (4.6.2) is equivalent to (4.6.1).
6.2. Small oscillations and the rigidity diagram. Consider a stable mechanical system with n degrees of freedom which oscillates near the equilibrium. Its potential energy is quadratic and determined by a symmetric positive definite matrix A. The squares of the fundamental frequencies of the system coincide with the eigenvalues X I , .. . , x, of this matrix. If a linear constraint is imposed on the system, the new fundamental frequencies will interlace with the original ones, where y l , . . . , y,-1 are the eigenvalues of the matrix A restricted to the corresponding hyperplane. The rectangular diagram wn describing (see Section 1.1) the separation of the squared frequencies of the linear system (4.6.4) under a linear constraint is called the rigidity diagram. Suppose now that the system (i.e., the matrix A) and the linear constraint (the hyperplane h c Rn) are chosen at random. It turns out that under broad conditions this random system is self-amenable (see [ 5 5 ] ) :as the number of degrees of freedom grows, the random rigidity diagram approaches (in an appropriate scaling) a nonrandom diagram. Let us show that this limiting diagram is exactly 0.
THEOREM Let. w, be the rigidity diagram of a random symmetric matrix A ( ~ ) of order r? with respect to a random hyperplane h c Rn. Assume that the normal to h is uniformly distributed o n the unit sphere i n Rn, and the matrix entries a (i y ) , i < j , are independent of h and among themselves, and are identically distributed ) 1. T h e n with mean value ~ ( a j ; ) )= 0 and variance ~ ( a ! ; ) =
uniformly i n u E R.
178
4.
PROOF.
YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS
Let us first average with respect t o the random constraint h .
L E M M A Let . Pn(x) = n ( x - xi) be the characteristic polynomial of a matrix A of order n, let h c Rn be a random hyperplane whose normal is uniformly distributed o n the unit sphere, and let Qn(x) = n ( x - yi) be the characteristic polynomial of the matrix Ah = pAp (where p is the orthogonal projection o n h ) . Then
PROOF.
Consider the partial fraction expansion of Qn/ Pn:
It is not difficult to see that pk = ti, where J = (I1,. . . , t n ) is the normal to h. Since the distribution of J is homogeneous, we have
because
C pk = 1. It remains to apply (4.6.3).
The theorem now follows from the semicircle law, stated as in (4.6.2). 6.3. The free convolution of probability distributions. In a series of papers and the monograph [167], D. Voiculescu constructed a deep analogue of probability theory which still has no generally accepted name. We will call it free probability the0ry.l It is based on an analogue of the ordinary convolution of two distributions p, v, which is called the free convolution [167] and is denoted by pmv. The problem leading to the definition of the free convolution can be stated as follows. Let G be the free group with two generators, and let Ul, U2 be the unitary operators in 12(G) corresponding to the left shifts by these generators. Denote by 5 = h9,, the vector from 12(G)determined by the 6-function supported by the unity of the group. Consider functions X I = cpl(Ul), X2 = cp2(U2)in the operators Ul, U2, and let p1, p2 be their spectral measures, i.e.,
for each polynomial f . As Voiculescu observed, the spectral measure of the sum X I + X 2 depends only on the spectral measures of the summands. This distribution p = p l El p2 is called the free convolution of the distributions p l , p2. Voiculescu [166] gave a purely analytical interpretation of the free convolution. The key notion here is the so-called R-transform. Following [166], denote by
the Cauchy-Stieltjes transform of a measure p which is a generating function of the moments m l , m2, . . . of this measure:
l Translation
Editor's Note. This name is now in general use
56. PLANCHEREL GROWTH AND SEMICIRCLE DIFFUSION
Then there exists a series
such that
The series (4.6.9) is called the R-transform of the measure p . With respect to the free convolution, it plays the role of the logarithm of the Fourier transform. Indeed, Voiculescu [I661 showed that the R-transform is additive with respect to the free convolution of measures:
Another useful fact from [I661 is that the identity (4.6.11) can be taken as an implicit definition of the free convolution. By analogy with ordinary probability theory, the coefficients rl ,r2,. . . of (4.6.9) could be called free semi-invariants. The first three of them coincide with the classical semi-invariants. The difference appears first for the coefficient r4, which differs from the classical semi-invariant
+
by the squared variance: r4 = s4 (m2 - m:)2. An essential difference of the free convolution from the ordinary one is its nonlinearity: the distributive law does not hold for B. Of course, the commutative and associative laws remain valid. An outstanding role in free probability theory is played by the semicircle distributions 1 (4.6.12) dpt(x) = -J a d x , 1x1 5 2t. 27rt It is easy to verify that the R-transform of these distributions is
It follows that
so that the distributions (4.6.12) form a semigroup with respect to the free convolution. Let us define the semicircle diffusion as the flow, in the space of measures M, determined by the formula
Voiculescu showed in the same paper [I661 that the Cauchy-Stieltjes transform
of the diffusing measure ut satisfies the Burgers equation
180
4.
YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
Since the transform (4.6.7) uniquely determines the measure p , we may say that the Burgers equation provides an adequate analytical description of the semicircle diffusion. 6.4. The semicircle diffusion and the Plancherel growth of Young diagrams. Consider a rectangular diagram w E Do with extrema a t points
and define its small perturbation wt, t > 0. By definition, the diagram wt E Do is obtained from w by attaching a small square of area vkt, where vk are the Plancherel transition probabilities
above each minimum xk. Thus (see Figure 13 on p. 54) the rectangular diagram wt has minima at the points (4.6.18)
X:
= xk
d~
a,
and maxima at the points yk, xk. Consider the moment generating function of the diagram wt. By the general formula (4.1.14), this rational fraction equals
PROPOSITION. The function R(x, t ) defined by (4.6.19) satisfies the identity
PROOF.Let us rewrite (4.6.19) in the following form:
Since
we obtain dR(x, 0) R(x, t) - R(x, 0) = -tR(x, 0)-dx and (4.6.20) follows in the limit as t -+ 0.
+ o(t),
Thus we have checked that if a rectangular diagram wt grows according to the Plancherel probabilities (4.6.17), then its transition distribution vt infinitesimally (for small t ) evolves according t o formula (4.6.14), i.e., diffuses by means of the free convolution with the semigroup (4.6.12). The link is provided by the Burgers equation. Let us consider more carefully the behaviour of the transition measure vt of the rectangular diagram wt. We will explicitly describe a discrete approximation fit
$6. PLANCHEREL GROWTH AND SEMICIRCLE DIFFUSION
181
of the measure ut, and show that these two measures coincide up to terms of order o ( t ) as t -+ 0. By definition, the measure fit attaches the weights
to the points x:
= xk
f
m,where
Ak=C--ui
,
k = I , . . . ,n.
iZk X k - X j
PROPOSITION. Denote by
the Cauchy-Stieltjes transfomn of the measure fit (recall that the corresponding transform of the transition measure ut of the diagram wt is given b y (4.6.19)). Then
PROOF. Let us rewrite the general summand in (4.6.23) as follows: -
X - X Z +-X - x i Uk
Ak
-
x
-
Xk
+
( X - x k ) Vk A k uk t (X -~ k - ~) k ~t
(G+ ( X -ukX I ) '
)t+
O(t))
It follows that
where
We have earlier seen that the function R ( x , t ) satisfies the analogous identity
where
(4.6.28) In order to verify that C = C , subtract from both sides the second summand of (4.6.26); we arrive at the formula
182
4.
YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS
It remains to prove (4.6.29). Substituting the values of Ak from (4.6.22), we can rewrite (4.6.29) in the form
Since --v j
xk
-
'/j Xj
-
( X - x k ) vj (xk- x j )( X -x j )
X -X j
(4.6.30) yields
Using the identity
we can finally rewrite (4.6.31) in the form
But in this form the formula is obvious, and the proof is complete. Formula (4.6.21) illustrates the character of mass transfer of the original discrete distribution v in the course of the semicircle diffusion during a small period of time t. It is worth mentioning that the semicircle diffusion has a number of differences from the ordinary Gaussian diffusion: ( 1 ) Each mass vk breaks into two almost equal point masses v:. ( 2 ) The latter masses disperse as xk f i.e., heavy particles disperse faster. ( 3 ) The barycentre ( v z x t v i x i ) / ( v z v i ) of the dispersing masses v f is of order Akt, and the moment is equal t o v k t ~ ( t ) . ( 4 ) The particles do not go to infinity within a finite time, i.e., the measures v t , fit are compactly supported.
+
a, +
+
6.5. The diffusion equation in Voiculescu's theory. Let us give a heuristic argument showing that the equation
can be regarded as the diffusion equation in free probability theory. Unfortunately, this equation looks more complicated than the ordinary one, because of the nonlinear character of free convolution. Let us compute the mean value of a test function f with respect to the discrete measure fi defined by (4.6.21) and (4.6.22). Since
57. ASYMPTOTIC SEPARATION OF ROOTS OF ORTHOGONAL POLYNOMIALS
we see that for t
=0
183
the derivative of the mean value equals
and we arrive at an integral sum for the integral in the right-hand side of (4.6.32). 57. Asymptotic separation of roots of orthogonal polynomials The main purpose of this section is to show that a wide class of orthogonal polynomials (including the classical polynomials of Jacobi, Hermite, and Laguerre) has a common universal asymptotics of mutual separation of roots. We emphasize that the question here is not in the distribution of roots, but in the character of their mutual separation. The section is organized as follows: we state the main result in Sections 7.1-7.2, and prove it in Sections 7.3-7.4. Section 7.5 contains some estimates which refine the main theorem in the case of the Chebyshev polynon~ials.
7.1. Separation of roots of orthogonal polynomials. Consider polynomials n
belonging to a family of polynomials orthogonal with respect to a certain measure dp(x). It is well-known ( [ 5 6 ] )that their roots satisfy the separation condition (4.1.2). For example, Table 1 presents the roots (for n = 15,16) of the Chebyshev polynomials of the second kind Un(x) =
+
sin (n 1) arccos x sin arccos x
1
the Hermite polynomials
and the Laguerre polynomials
Figure 11 (on p. 47) displays the corresponding rectangular diagrams. We see that the diagrams of root separation in this figure are strikingly alike (up t o scale parameters). To emphasize this fact, we have laid the (appropriately scaled) graph of the function
upon these diagrams. Note that the area A v < Q(u)) of this function is equal to one.
=
A(R) of the subgraph {(u, v): lul 5
The function R appeared first in [12]and [I381 in connection with a problem of the asymptotic representation theory of the symmetric group (see 52 in Chapter 3 for the description of the limit shape of large Young diagrams). We will show that
184
4.
YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS
-
-
-
TABLE1 The roots of polynomials (of degrees 16 and 15) Hermite Laguerre Chebyshev
this function describes the asymptotics of the root separation for a wide class of orthogonal polynomials. 7.2. Linear recurrence relations. I11 what follows it will be more convenient t o use another standard approach to the theory of orthogonal polynomials when they arise as solutions of difference equations (see, e.g., [2]). Consider a second order linear difference equation
with spectral parameter u, and let 4, = P n ( u ) , n = 0,1,2, . . . , be the unique solution of this equation satisfying the initial conditions
Obviously, Pn(u) is a monic polynomial of degree n. Assuming that c i > 0 for all n 2 1, one can easily see that the roots of the polynomials Pn(u) are real and simple, and the roots of two adjacent polynomials Pn-l(u), Pn(u) interlace (see [2], Chapter 4). Favard's theorem (see [52], Chapter 8.6) guarantees that there exists a measure p for which the polynomials Pn are orthogonal: 00
(4.7.4)
P,(u) Pn(u)dp(u) = 0 f o r m
# n.
Conversely, the polynomials orthogonal with respect to a certain measure satisfy a linear recurrence relation of the form (4.7.2). Consider the family of orthogonal polynomials { P , ( U ) ) ~ =determined ~ by the linear recurrence relation
57. ASYMPTOTIC SEPARATION O F ROOTS O F ORTHOGONAL POLYNOMIALS
185
It is well-known that the roots of the adjacent polynomials
interlace: x1 < yl < x2 < . . . < xn-1 < yn-1 < x,. Let us associate with the pair of polynomials (4.7.6) a continuous piecewise linear function v = w n ( u ) (we call it a diagram) such that
Pn-1 ( u ) wk ( u ) = sign pn(u) ' w n ( u ) = Ju- znI for sufficiently large ( u ( , where zn = C x i - C yi. The roots of Pn are the minima of w,, and the roots of Pn-l, its maxima (Figure 21, on p. 145). The key result of this section is the following theorem.
THEOREM Let. P n ( u ) be the orthogonal polynomials determined by equation (4.7.2) with initial conditions (4.7.3), and let wn be the rectangular diagram describing the mutual separation of roots of the polynomials P n - l ( u ) , P n ( u ) . Assume that the coeficients of (4.7.2) satisfy the following conditions: Cn-1
lim -= 1, cn
(4.7.8)
n+m
bn+l - bn cn
lim
n-m
= 0.
Then lim
n-cc
1 cn
-W,(UC~-~
+ bn) = O ( u )
~lniformlyi n u E R, where the function R is defined by (4.7.1). In [I181this result was established with less generality: instead of conditions (4.7.8) it was assumed that
(4.7.10)
lim c n = c > O ,
n-00
lim bn=O.
n-cc
Let d p be the measure for which the family {P,) is orthogonal. For condition (4.7.10) to be satisfied, it is sufficient that d p ( u ) be supported by the interval [ - I l l ] and the function log pf(cos0) be integrable on the interval [0,T]. On the other hand, the theorem certainly fails if the support of d p is not connected. The classical Hermite and Laguerre polynomials are in an intermediate position: they satisfy (4.7.8), but not (4.7.10). It is known [56]that if c = 1 in (4.7.10), then there exists a common limiting density of the distribution of roots: lim
n+cc
jul
n.
I 1.
The roots of Hermite polynomials have a different limiting density:
1 Xi lim -#{i: - < U ) = n-cc n
6
2T
-1
JTZ?dxl
1111
< 11
but the asymptotics of mutual separation of roots is the same in both cases. Thus our result is more universal (but also coarser) than the theorems on the limiting distribution of roots of orthogonal polynomials.
186
4.
YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS
7.3. T h e e q u a t i o n for t h e limiting R-function. Let Rn (u) = Pn- 1(u)/Pn (u) be the R-function of the diagram w,. Dividing both sides of the identity Pn+l(u) = ( U
-
bn+l)Pn(u) - c i ~ n - l ( u )
by Pn(u), we obtain a recurrence relation for the R-functions:
Using the formulas from Section 2.5, we go over to the normalized functions, i.e., the R-functions
+
of the diagrams Gn(u) = wn(ucn-l bn) / cn-1, Gn+i (u) = wn+l(ucn + bn+l) / cn. After we replace u by (u - bn+l)/cn, formula (4.7.11) takes the form
where A, = bn+l - b,. Assume that the limit
exists for sufficiently large 1ul. Then in view of (4.7.8) we obtain the equation
from which it follows that R ( U ) = i ( u - d m ) = Rn(u) (the sign before the square root is chosen from the condition ~ ( m=) lim ~ , ( m )= 0). In order to prove that the limit (4.7.12) exists, we will consider the behaviour of the Taylor coefficients at infinity.
7.4. T h e Taylor coefficients. Let us rewrite (4.7.11) in the form
and consider the power series
where
57. ASYMPTOTIC SEPARATION OF ROOTS OF ORTHOGONAL POLYNOMIALS
187
we obtain recurrence relations for the coefficients:
for k
= 0,1,2,
More generally, (4.7.14) and (4.7.15) imply LEMMA.The coefficients r P + ' ) are homogeneous polynomials of degree k in the first k - 1 variables of the sequence c,, A,, c,-1, An-l,. . . . COROLLARY.Under the assumptions of the theorem, the limits lim,,, fP)= Ck exist, where f r ) = r?)/ck are the Taylor coefficients of the normalized R function
&
(2kk) are the Catalan numbers. Note that the limiting coefficients C2k = These numbers also arise as the coefficients of the expansion Rn(u) = Co/u C2/u3 + . . . of the R-function of the diagram R. The convergence of R-functions lim R, (u)= Rn (u) implies the uniform convergence of diagrams lim Gn(u) = R (u), and the theorem follows.
+
7.5. Separation of roots of the Chebyshev polynomials. In this section we give a direct proof of Theorem 7.2 in the special case of the Chebyshev polynomials of the second kind. The proof contains a simple bound on the rate of convergence in (4.7.9).
s,
PROPOSITION.Let xk = - cos k = 1,.. . , n , be the roots of the Chebyshev polynomials of the second kind U, (x), and let yk = - cos $, k = 1, . . . , n- 1, be the roots of Un-l(x). Denote by w, the rectangular diagram describing the interlacing sequences {xk), {yk). Then (4.7.16) as n
I ~ n ( u) R(u)I
--, co,uniformly
= O(l/n)
in u E R.
PROOF. It suffices t o prove (4.7.16) for u the definition (4.7.1) that
= xk
and for u
= yk.
It is clear from
By the description (4.1.4) of the diagram w, we have k
w ( Y ~=) 2 ~ k- (yk - Z )
=
-yk
+ 2~~,
where zk
=x
( y i - xi).
i= 1
L
U!S
"41+ YZ)U!S
-
T+"D
U!S
+
= YZZ
TfUo(l ~Z)U!S
q n u n o j ay? Lq sau!sor,
wyq aas am
JO
urns ayl 8u!Ljgdur!s
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Comments on Kerov's Thesis G. OLSHANSKI Chapter 1 1) Section 1, The ergodic method. In connection with the ergodic theorem proved in the text, see also Olshanski and Vershik [A.46] and Olshanski [A.44], Proposition 10.8. 2) Section 3, Theorem 1 (Boundary of Kingman's graph). Both Kingman's theorem and Thoma's theorem are particular cases of a more general result proved by Kerov, Okounkov, and Olshanski [A.34].
Chapter 2 3) Section 2.2, Asymptotics of the number of skew Young diagrams. Detailed proofs and further results- are contained in Wassermann [A.55], in Okounkov and Olshanski [A.41], and in Olshanski, Regev, and Vershik [A.45].
4) Section 2.3, Thoma's theorem. A generalization of Thoma's theorem is given by Kerov, Okounkov, and 01shanski [A.34]. The proof is based on Vershik and Kerov's asymptotic method, but does not use the multiplicativity property.
5) Section 9.3, The conjecture. One more case when the Kerov conjecture is known to be true is that of Jack polynomials, which are a degenerate case of Macdonald's polynomials Px(x; q, t ) (t = q1Ia, q -4 1, a > 0 is fixed). See Kerov, Okounkov, and Olshanski [A.34]. In the general case, the conjecture remains open, and any advance in this direction would be very interesting. In particular, the case of Hall-Littlewood polynomials (q = 0) is interesting because of connections with the results of Kerov and Vershik [A.54] on the characters of certain infinite matrix groups over finite fields. Chapter 3 6) Section 1.4, The Ulam problem. A new achievement related to the Ulam problem is due t o Baik, Deift, and Johansson [A.3]: they found the asymptotics of the distribution of 7-1(A) (the length of the first row of the random Plancherel diagram). It turned out that the result coincides with the asymptotic distribution for the largest eigenvalue of the random Hermitian N x N matrix with Gaussian measure, N -+ co. Further results in this direction: Baik, Deift, and Johansson [A.4], Okounkov [A.37], Borodin,
198
COMMENTS ON KEROV'S THESIS
Okounkov, and Olshanski [A.11], and Johansson [A.30]; see also the expository paper by Deift [A.15].
7) Section 2, Gaussian limit of the Plancherel measure. In the late 90s, Kerov returned to this subject and found another approach to the results stated in Section 2.1. His proof was reconstructed by Ivanov and 01shanski [A.28]. As shown in that paper, Kerov's method also allows one to describe the fluctuations of the transition measures of the random Plancherel diagrams; the result shows a striking similarity with the central limit theorem for random matrices (Diaconis and Shahshahani [A.161, and Johansson [A.29]). Kerov's new approach uses the results of his joint paper with Ivanov [A.27], which contains a detailed proof of Proposition 1 and further results about convolution of conjugacy classes in symmetric groups. The algebra of functions on Young diagrams A, briefly mentioned in the very beginning of Section 2.3, is an important object; in more detail it is considered in Kerov and Olshanski [124], see also Lascoux and Thibon [A.36]. This algebra (which is denoted in [A.28] as A) can be identified with the algebra of shifted symmetric functions in the row coordinates of X (see Okounkov and Olshanski [A.41]). Using the modified Frobenius coordinates of A, the same algebra A can also be identified with the algebra of symmetric functions in its "super" realization (see Olshanski, Regev, and Vershik [A.45]). Kerov's approach is largely based on exploiting relations between various natural bases and systems of generators in the algebra A. Here there are a number of open questions, see, e.g., Biane [A.7], [A.8]. An alternative proof of Theorem 1 is given by Hora [A.25]. An analog of Theorem 1 for characters of projective representations of symmetric groups was obtained by Ivanov [A.26]; his paper incorporates a "projective" version of the results of [A.27].
8) Section 3, Distribution of symmetry types for tensors of large degree. , ~ regime, when Biane studied the asymptotics of the measure p ~ in another m const N ' / ~ Then . a different limit shape arises, see [A.6], [A.7], [A.8]. 9) Theorem 2 (Section 3.4) was rediscovered by Tracy and Widom [A.50] and by Johansson [A.30].
Chapter 4 10) Kerov's ideas related to the concept of continual diagrams and their transition measures were further developed and exploited by Biane, see [A.5], [A.9]. 11) The material of Sections 1-3 is discussed in more detail and greater generality in Kerov's paper [A.33]. Further results concerning the hook walk algorithm of Section 4 were obtained by Romik [A.47].
Additional References A.1. D. Aldous and P. Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deifl-Johansson theorem, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 413-432. A.2. M. Adler and P. van Moerbeke, Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Comm. Pure Appl. Math. 54 (2001), 153-205. A.3. J. Baik, P. Deift, and K. Johansson, O n the distribution of the length of the longest increasing subsequence of random permutations, J . Amer. Math. Soc. 12 (1999), 1119-1178. O n the distribution of the length of the second row of a Young diagram under A.4. , Plancherel measure, Geom. Funct. Anal. 10 (2000), 702-731. A.5. P. Biane, Representations of symmetric groups and free probability, Adv. Math. 138 (1998), 126-181. A.6. , Approximate factorization and concentration for characters of symmetric groups, Internat. Math. Res. Notices 2001, 179-192. A.7. , Free cumulants and representations of large symmetric groups, XIIIth Internat. Congr. Math. Phys. (London, 2000), International Press, Boston, MA, 2001, pp. 321-326. A.8. , Free cumulants and characters of symmetric groups, Preprint, 2001. Free probability and combinatorics, Proc. Internat. Congr. Math. (Beijing, 2002), A.9. , Vol. 11, World Sci. Publ., Singapore, 2002. Characters of symmetric groups and free cumulants, in [A.56],pp. 185-200. A.lO. , A . l l . A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J . Amer. Math. Soc. 13 (2000), 481-515. A.12. A. Borodin and G. Olshanski, Infinite random matrices and ergodic measures, Comm. Math. Phys. 223 (2001), 87-123. z-measures o n partitions, Robinson-Schensted-Knuth correspondence, and P = 2 A.13. , random matrix ensembles, Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ., vol. 40, Cambridge Univ. Press, Cambridge, 2001, pp. 71-94. A.14. M. Bozejko, B. Kiimmerer, and R. Speicher, q-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), 129-154. A.15. P. Deift, Integrable systems and combinatorial theory, Notices Amer. Math. Soc. 47 (2000), 631-640. A.16. P. Diaconis and M. Shahshahani, O n the eigenvalues of random matrices, J. Appl. Probab. 31A (1994), 49-62. A.17. J. Fulman, A probabilistic approach toward conjugacy classes i n the finite general linear group, J . Algebra 212 (1999), 557-590. A.18. , New examples of potential theory o n Bratteli diagrams, Preprint, 1999, arXiv: math.C0/9912148. A.19. A. Gnedin, The representation of composition structures, Ann. Probab. 25 (1997), 14371450. Three sampling formulas, Preprint, 2002, arXiv:math.PR/0210319. A.20. , A.21. A. Gnedin and S. Kerov, The Plancherel measure of the Young-Fibonacci graph, Math. Proc. Cambridge Philos. Soc. 129 (2000), 433-446. A.22. , A characterization of GEM distributions, Combin. Probab. Comput. 10 (2001), 213-217. A.23. , Fibonacci solitaire, Random Structures Algorithms 20 (2002), 71-88. A.24. F. Goodman and S. Kerov, The Martin boundary of the Young-Fibonacci lattice, J . Algebraic Corrtbin. 11 (2000), 17-48. A.25. A. Hora, Central limit theorem for the adjacency operators o n the infinite symmetric group, Comm. Math. Phys. 195 (1998), 4055416.
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A.26. V . N. Ivanov, Gaussian limit for projective characters of large symmetric groups, Zapiski Nauchn. Semin. POMI 283 (2001),73-97; English transl., t o appear in J . Math. Sci. (New York). A.27. V . Ivanov and S. Kerov, The algebra of conjugacy classes i n symmetric groups, and partial permutations, Zapiski Nauchn. Semin. POMI 256 (1999),95-120; English transl., J . Math. Sci. (New Y o r k ) 107 (2001),4212-4230. A.28. V . Ivanov and G . Olshanski, Kerov's central limit theorem for the Plancherel measure on Young diagrams, Symmetric Functions 2001: Surveys o f Developments and Perspectives ( S . Fomin, ed.), Kluwer, Dordrecht, 2002, pp. 93-151. A.29. K . Johansson, O n jluctuations of eigenvalues of random Hermatian matrices, Duke Math. J. 91 (1998), 151-204. A.30. , Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. o f Math. ( 2 ) 153 (2001),259-296. A.31. S. Kerov, Double function algebras on a finite group, Zap. Nauchn. Sem. LOMI 39 (1974), 182-185; English transl., J . Soviet Math. 8 (1977), 136-139. A.32. , Duality of finite-dimensional *-algebras, Vestnik Leningrad. Univ. 1974,no. 7 (Ser. Mat. Mekh. A s h . vyp. 2 ) , 23-29; English transl., Vestnik Leningrad Univ. Math. 7 (1979), 122-130. A.33. , Interlacing measures, Kirillov's Seminar on Representation Theory, Amer. Math. Soc. Transl. ( 2 ) 181 (1998),35-83. A.34. S. Kerov, A . Okounkov, and G . Olshanski, The boundary of Young graph with Jack edge multiplicities, Internat. Math. Res. Notices 1998,173-199. A.35. S. V . Kerov and N. V . Tsilevich, The Markov-Krein correspondence i n several dimensions, Zapiski Nauchn. Semin. POMI 283 (2001),98-122; English transl., t o appear in J . Math. Sci. (New Y o r k ) . A.36. A . Lascoux and J.-Y. Thibon, Vertex operators and the class algebras of the symmetric groups, Zapiski Nauchn. Semin. POMI 283 (2001), 156-177; English transl., t o appear in J . Math. Sci. ( N e w York).. A.37. A . Okounkov, Random matrices and random permutations, Internat. Math. Res. Notices 2000,,1043-1095. A.38. -, Infinite wedge and random partitions, Selecta Math. (N.S.) 7 (2001), 1-25. A.39. -, S L ( 2 ) and z-measures, Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ., vol. 40, Cambridge Univ. Press, Cambridge, 2001, pp. 407-420. A.40. , Random trees and moduli of curves, in [A.56], pp. 89-126. A.41. A . Okounkov and G . Olshanski, Shijled Schur functions, Algebra i Analiz 9 (1997), no. 2, 73-146; English transl., St. Petersburg Math. J . 9 (1998),239-300. A.42. A . Okounkov and R . Pandharipande, Gromov- Witten theory, Hunuitz theory, and completed cycles, Preprint, 2002, arXiv :math.AG/0204305. A.43. , The equivariant Gromov-Witten theory of P1,Preprint, 2002, arXiv:math.AG/ 0207233. A.44. G.Olshanski, The problem of harmonic analysis on the infinite-dimensional unitary group, t o appear in J . Funct. Anal.; available via arXiv:math.RT/0109193. A.45. G . Olshanski, A . Regev, and A . Vershik, Frobenius-Schur functions, Studies in Memory o f Issai Schur ( A . Joseph et al., eds.), Progr. Math., vol. 210, Birkhauser, Basel, 2003, pp. 251-300. A.46. G . Olshanski and A . Vershik, Ergodic unitanly invariant measures on the space of infinite Hermitian matrices, Contemporary Mathematical Physics: F. A . Berezin Memorial Volume, Amer. Math. Soc. Transl. ( 2 ) 175 (1996), 137-175. A.47. D. Romik, Explicit formulas for hook walks on continual Young diagrams, Preprint. A.48. R . Speicher, Combinatorial theory of the free product with amalgamation and operatorvalued free probability theory, Mem. Amer. Math. Soc. 132 (1998),no. 627. pp. 53-74. A.49. -, Free probability theory and random matrices, in [A.56], A.50. C. A . Tracy and H . W i d o m , O n the distnbutions of the lengths of the longest monotone subsequences i n random words, Probab. Theory Related Fields 119 (2001),350-380. A.51. N . Tsilevich, Distribution of mean values of some random matrices, Zapiski Nauchn. Semin. POMI 240 (1997),268-279; English transl., J . Math. Sci. (New Y o r k ) 96 (1999),3616-3623.
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A.52. A. M. Vershik, Statistical mechanics of combinatorial partitions, and their limit configurations., Funkts. Anal. i Prilozhen. 30 (1996), no. 2, 19-30; English transl., Funct. Anal. Appl. 30 (1996), 90-105. A.53. , Two lectures o n the asymptotic representation theory and statistics of Young diagrams, in [A.56], pp. 161-182. A.54. A. M. Vershik and S. V.Kerov, O n a n infinite-dimensiond group over a jinite field, Funkts. Anal. i Prilozhen. 32 (1998), no. 3, 3-10; English transl., Funct. Anal. Appl. 32 (1998), 147-152. A.55. A. J. Wassermann, Automorphic Actions of Compact Groups o n Operator Algebras, Ph.D. Thesis, University of Pennsylvania, 1981. A.56. A. M. Vershik (Editor), Asymptotic Combinatorics with Applications t o Mathematical Physics, Springer-Verlag, Berlin, 2003.
Titles in This Series S. V. Kerov, Asymptotic representation theory of the symmetric group and its applications to analysis, 2003 Kenji Ueno, Algebraic geometry 3: Further study of schemes, 2003 Masaki Kashiwara, D-modules and microlocal calculus, 2003 G. V. Badalyan, Quasipower series and quasianalytic classes of functions, 2002 Tatsuo Kimura, Introduction t o prehomogeneous vector spaces, 2003 L. S. Grinblat, Algebras of sets and combinatorics, 2002 V. N. Sachkov and V. E. Tarakanov, Combinatorics of nonnegative matrices, 2002 A. V. Mel'nikov, S. N. Volkov, and M. L. Nechaev, Mathematics of financial obligations, 2002 Takeo Ohsawa, Analysis of several complex variables, 2002 Toshitake Kohno, Conformal field theory and topology, 2002 Yasumasa Nishiura, Far-from-equilibrium dynamics, 2002 Yukio Matsumoto, An introduction to Morse theory, 2002 Ken'ichi Ohshika, Discrete groups, 2002 Yuji Shimizu and Kenji Ueno, Advances in moduli theory, 2002 Seiki Nishikawa, Variational problems in geometry, 2001 A. M. Vinogradov, Cohomological analysis of partial differential equations and Secondary Calculus, 2001 Te Sun Han and Kingo Kobayashi, Mathematics of information and coding, 2002 V. P. Maslov and G. A. Omel'yanov, Geometric asymptotics for nonlinear PDE. 1: 2001 Shigeyuki Morita, Geometry of differential forms, 2001 V. V. Prasolov and V. M. Tikhomirov, Geometry, 2001 Shigeyuki Morita, Geometry of charact.eristic classes, 2001 V. A. Smirnov, Simplicia1 and operad methods in algebraic topology, 2001 Kenji Ueno, Algebraic geometry 2: Sheaves and cohomology, 2001 Yu. N. Lin'kov, Asymptotic statistical methods for stochastic processes, 2001 Minoru Wakimoto, Infinite-dimensional Lie algebras, 2001 Valery B. Nevzorov, Records: Mathematical theory, 2001 Toshio Nishino, Function theory in several complex variables, 2001 Yu. P. Solovyov and E. V. Troitsky, CL-algebras and elliptic operators in differential topology, 2001 Shun-ichi Amari and Hiroshi Nagaoka, Methods of information geometry, 2000 Alexander N. Starkov, Dynamical systems on homogeneous spaces, 2000 Mitsuru Ikawa, Hyperbolic partial differential equations and wave phenomena, 2000 V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, 2000 A. V. Fursikov, Optimal control of distributed systems. Theory and applications, 2000 Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Number theory 1: Fermat's dream, 2000 Kenji Ueno, Algebraic Geometry 1: From algebraic varieties to schemes, 1999 A. V. Mel'nikov, Financial markets, 1999 Hajime Sato, Algebraic topology: an intuitive approach, 1999 I. S. Krasil'shchik and A. M. Vinogradov, Editors, Symmetries and conservation laws for differential equations of mathematical physics, 1999 Ya. G. Berkovich and E. M. Zhmud', Characters of finite groups. Part 2, 1999
TITLES IN THIS SERIES
A. A. Milyutin and N. P. Osmolovskii, Calculus of variations and optimal control, 1998 V. E. Voskresenskii, Algebraic groups and their birational invariants, 1998 Mitsuo Morimoto, Analytic functionals on the sphere, 1998 Satoru Igari, Real analysis-with an introduction to wavelet theory, 1998 L. M. Lerman and Ya. L. Umanskiy, Four-dimensional integrable Hamiltonian systems with simple singular points (topological aspects), 1998 S. K. Godunov, Modern aspects of linear algebra, 1998 Ya-Zhe Chen and Lan-Cheng Wu, Second order elliptic equations and elliptic systems, 1998 Yu. A. Davydov, M. A. Lifshits, and N. V. Smorodina, Local properties of distributions of stochastic functionals, 1998 Ya. G. Berkovich and E. M. Zhmud', Characters of finite groups. Part 1, 1998 E. M. Landis, Second order equations of elliptic and parabolic type, 1998 Viktor Prasolov and Yuri Solovyev, Elliptic functions and elliptic integrals, 1997 S. K. Godunov, Ordinary differential equations with constant coefficient, 1997 Junjiro Noguchi, Introduction to complex analysis, 1998 Masaya Yamaguti, Masayoshi Hata, and J u n Kigami, Mathematics of fractals, 1997 Kenji Ueno, An introduction to algebraic geometry, 1997 V. V. Ishkhanov, B. B. Lur'e, and D. K. Faddeev, The embedding problem in Galois theory, 1997 E. I. Gordon, Nonstandard methods in commutative harmonic analysis, 1997 A. Ya. Dorogovtsev, D. S. Silvestrov, A. V. Skorokhod, and M. I. Yadrenko, Probability theory: Collection of problems, 1997 M. V. Boldin, G. I. Simonova, and Yu. N. Tyurin, Sign-based methods in linear statistical models, 1997 Michael Blank, Discreteness and continuity in problems of chaotic dynamics, 1997 V. G. Osmolovskii, Linear and nonlinear perturbations of the operator div, 1997 S. Ya. Khavinson, Best approximation by linear superpositions (approximate nomography), 1997 Hideki Omori, Infinite-dimensional Lie groups, 1997 V. B. Kolmanovskii and L. E. Shaikhet, Control of systems with aftereffect, 1996 V. N. Shevchenko, Qualitative topics in integer linear programming, 1997 Yu. Safarov and D. Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, 1997 V. V. Prasolov and A. B. Sossinsky, Knots, links, braids and 3-manifolds. An introduction t o the new invariants in low-dimensional topology, 1997 S. Kh. Aranson, G. R. Belitsky, and E. V. Zhuzhoma, Introduction to the qualitative theory of dynamical systems on surfaces, 1996 R. S. Ismagilov, Representations of infinite-dimensional groups, 1996 S. Yu. Slavyanov, Asymptotic solutions of the one-dimensional Schrodinger equation, 1996 B. Ya. Levin, Lectures on entire functions, 1996 Takashi Sakai, Riemannian geometry, 1996
For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/.
Asymptotic representation theory of symmetric groups deals with two types of problems: asymptotic properties of representations of symmetric groups of large order, and representations of the limiting object, i.e., the infinite symmetric group. The author contributed significantly in the development of problems of both types, and his book presents an account of these contributions, as well as those of other researchers. Among the problems of the first type, the author discusses the properties of the distribution of the normalized cycle length in a random permutation, and the limiting shape of a random (with respect to the Plancherel measure) Young diagram. He also studies stochastic properties of the deviations of random diagrams from the limiting curve. Among the problems of the second type, the author studies an important problem of computing irreducible characters of the infinite symmetric group. This leads him to the study of a continuous analog of the notion of Young diagram, and, in particular, to a continuous analogue of the hook walk algorithm, which is well known in the combinatorics of finite Young diagrams. In turn, this construction provides a completely new description of the relation between the classical moment problems of A Hausdorff and Markov.
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EDITORIAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. Margulis James D. Stasheff (Chair) ASL Subcommittee Steffen Lempp (Chair) IMS Subcommittee Mark I. Freidlin (Chair)
Tkanslated from t h e Russian manuscript by N. V. Tsilevich 2000 Mathematics Subject Classijication. P r i m a r y 20C30, 20P05, 22D10.
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Library of Congress Cataloging-in-Publication Data Kerov, S. V. (Sergei Vasil'evich), 1946-2000. [Asimptoticheskaia teoriia predstavleniia simmetricheskoi gruppy i ee primeneniia v analize. English] Asymptotic representation theory of the symmetric group and its applications in analysis / S. V. Kerov ; translated by N. Tsilevich. p. cm. - (Translations of mathematical monographs, ISSN 0065-9282 ; v. 219) Includes bibliographical references. ISBN 0-8218-3440-1 (acid-free paper) 1. Symmetry groups-Asymptotic theory. 2. Representations of groups. I. Title. 11. Series.
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