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Tables of the SU(mn) 2 SU(m) x SU(n) Coefficients of Fractional Parentage
Authors
Jin-Quan Chen Xiong-Biao Wu Mei-Juan Gao
\
World Scientific Singapore New Jersey Lon-ong
Publirhcd by
World Scientific Publishing Co.Pte. Ltd. P 0 Box 128. Famr Road, Singapore 9128 USA o f i e : 687 Hattwell Street. Teaneck, NJ 07666 UK e e : 73 Lynton Meed, Toaeridge. London N20 8DH
Library of Congress Cataloging-in-Publication data is available.
TABLES OF THE SU(m) 3 SWm) x SU(n) C O m C E N l S OF FRACTIONAL PARENTAGE Capy'ight O 1991 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, orports thereof, may not be reproduced in anyform or by any meam, electronicormcchnical, including pharocopying, recording orany information storage and retrieval system now bwwn or to be invented, without wrillen permirsion from the Publisher.
ISBN 981-02-0113-3
Printed in Singapore by Utopia Press.
Preface
Since the advent of the Racah CFP (coefficients of fractional parentage) technique in atomic spectroscopy (Ra-42, -43, -49, -51), it has been acquiring an ever increasing importance in physics. Of the various kinds of CFP, the SU(mn) > SU(m) x SU(n) holds a central position. It is due to the fact that if a particle has ni single-particle states in the subspace V;., one can use the SU(nlnzn3 . . .) > SU(n1) x SU(nz)x SU(n3). . . irreducible basis as the starting point for a manyparticle wave function. For instance, in the nuclear shell model, one uses the SU(4(21+ 1)) > SU(2l+l) xSU(4)(> SU(2) x SU(2)) basis; in hypernuclei, the SU(6(21+1)) > SU(2l+l) x SU(6)(> SU(3) x SU(2)) basis, while in the quark model of nuclei, if a quark has three colors and nf flavors, one uses the SU(6nf(21 1)) > SU(21+ 1) x SU(6nf)(> SU(3) x SU(2nf)(> SU(2) x SU(nf)) basis. On account of the Racah factorization lemma ( h 5 1 ) , the CFP for the above bases always involves the SU(mn) > SU(m) x SU(n) CFP. For example, for the last case, it involves the CFP of SU(6nj(21 1)) > SU(6nf) x SU(21+ I), SU(6nf) > SU(3) x SU(2nf) and SU(2nf) > SU(2) x SU(n/). At first sight it seems that we need lots of SU(mn) > SU(m) x SU(n) CFP tables for all possible values of m and n. The SU(4) > SU(2) x SU(2) CFP have been known for a long time in nuclear physics (numerical table in Ja-51 and El-53, and analytic expressions for multiplicity free coefficients in He-69). In the sixties, the SU(6) > SU(3) x SU(2) CFP are badly needed in particle physics related to the three flavors of quarks and in hypernuclear physics related to the three baryons N, P and A, and there exist a lot of scattered results on the SU(6) > SU(3) x SU(2) CFP (Ca-65, Co-65, Sc-65, Ma-76, Zh-77, So-79,) St-79a, Da-81, Ha-81, Bi-82, So-84), among which only So-79 and So-84 offered systematic tables. Due to the fact that the labor involved in the calculation of the SU(mn) CFP increases drastically with m and n, no SU(mn) > SU(m) x SU(n) CFP existed before 1984 for higher m and n except for a special (in fact a trivial) case of the SU(12) > SU(4) x SU(3) CFP due to Ma-78. However, it is now realized (Ch-8la, Ob-82) that the SU(mn) > SU(m) x SU(n) f2-particle CFP are precisely the isoscalar factors for the permutation group chain S(f1 f2) > S(f1) x S(fz), and thus the SU(mn) > SU(m) x SU(n) CFP can be calculated via the permutation goup coefficients and their values do not depend on m and n explicitly. It is this breakthrough in the field of group theory which makes it possible to calculate and tabulate the SU(mn) CFP in a rank independent way instead of one m and one n at a time. The SU(mn) > SU(m) x SU(n) one-body CFP have been calculated and tabulated in this way (Ch-84a). In this book the SU(mn) > SU(m) x SU(n)fz-particle CFP, i.e. the S(f) > S(fl) x S
+
+
+
vi
Tables of the SU(rnn)~SU(rn)xSU(n)Coeficients of Fractional Parentage
(f2) isoscalar factors, for f2 = 1,2,3,4,5 and for systems with up to six particles are computed from the Clebsch-Gordan coefficients(Ga-85) and the S(f) 1S(fi) x S(f2) subduction coefficients (Ch-83b) of the permutation groups. Therefore it is an extension of the tables in Ch-84a. TO tabulate the SU(mn) CFP in a rank independent way, the partitions are used as the irreducible representation labels of unitary groups instead of the ad hoc use of dimensions in particle physics or other quantum numbers depending on the particular rn and n (such as the spin S and isospin T for m = n = 2). All the coefficientsare in the form of square roots of simple fractions. The development of the program constitutes the main part of the Master thesis "A Rank Independent Calculation of the SU(mn) > SU(m) x SU(n) Coefficients of Fractional Parentage" by Xiong-Biao Wu at the Department of Physics, Nanjing University. The program was written in Fortran-77 and implemented on IBM PC/AT computer. We owe our thanks to Professors Yi-Jin Shi, Michel Vallieres, Da Hsuan Feng and Fan Wang for their valuable discussions, and to Bing-Qing Chen for preparing the final output of all the tables.
December 1988, Nanjing University and Drexel University Jan-Quan Chen Xiong-Biao Wu Mei-Juan Gao
Contents
Preface
I.
The Algorithm
1. 2. 3. 4. 5. 6. 7. 8. 9.
The SU(rnn) > SU(m) x SU(n) CFP The CG Coefficients of Permutation Groups The S(f) > S(fi) x S(fi) ISF The Calculation Formula for the CFP The Properties of the CFP The Phase Convention The Evaluation of the CFP Discussion Appendix
II. III.
The Use of the Tables Tables
References
I. The Algorithm
> SU(m) SU(m) x SU(n) CFP is given in detail in Sec. Sec. 7.16 7.16 of the The theory of the SU(mn) :::> monograph Ch-89a, Ch-89a, as well as in Ch-81a, Ch-8la, Ch-83a and Ch-84a. In the following following we only give the key equations involved in the calculation calculation and leave out all the proofs. 1. The The SU(mn):::> SU(mn)> SU(m)xSU(n) SU(m)xSU(n) CFP CFP 1. Let
[II']
,8'[u']W{lJl']W~
I
)
,
[II"]
,8"[u"]W{'lJl"]W~'
I
)
(1)
,
+
be the SU(mn) SU(mn) :::> II), (fi (II + 1, > SU(m) SU(m) x SU(n) irreducible bases for the particles (1, (1,.. .. .. ,,fl), 1 ,... . . . ff)) and (1, /2, respectively, (1,.. .. .. ,, f) f ) with ff = = II fi + f2, respectively, where [u], [a], lJl] b ] and [II] [v] are partition labels for the irreps (irreducible SU(m), SU(n) and SU(mn) respectively, respectively, W W1(W2) (irreducible representations) of SU(m),SU(n) 1 (W2 ) is the component index for the irrep [u]( lJ.t]), and PI, f3', f3" [a](b]), P" and f3/3 are inner multiplicity multiplicity labels, labels, a' 1 2 , ... (" ') f3" = 1,2, .. . (-",,"11") , f3= 1,2, . .. , (UJ.lII) . 1-'=, UJ.lII, Vr
(2)
The integers J.I'II') etc. are determined (alp'vl) determined by the Clebsch-Gordan Clebsch-Gordan (CG) series series of the permutation integers (u' group fi-particle CFP are defined as the coefficients coefficients in the following following expansion (for group (It-66). The f2-particle brackets around u,J.I notational convenience, convenience, we dropped the square brackets a,p etc.) etc.):: (3a)
¢J and where B, 0,4
are the outer multiplicity labels, labels,
T T
B=1,2, . . . {u'u"u},
¢J=1,2, . . . {J.I'J.I"J.I},
T=1,2, . .. {II'II"II},
(4)
and the integers {u' { ~u"' u"} a ~ etc. ~ a are ~ ~decided } by the Littlewood rule; rule; the square square bracket denotes denotes that the bases are to be combined into the irreducible basis [U]Wl and lJ.t]W in terms of the CG [a]Wl b]Wz2 irreducible coefficients coefficients of SU(m) and SU(n) SU(n) respectively
[I
[II'] ) ,8'U'J.I'
I
[II"] )] [<1]9[1']
"
L..-i
W'W'W"W/I 1
2
1
:iii
C[<1]9.W. C[I']
I,8'u'W'[II']J.I'W' ) I,8" u"W" [II"] ) J.I"W" 1
2
1
2
.
(3b)
22
Tables Parentage Tablesofofthe theSU(mn)::>SU(m)xSU(n) SU(mn)3SU(m)xSU(n) Coefficients CoeficientaofofFractional F~actional Parentage
Both Boththe theinnerinner-and andouter-multiplicities outer-multiplicities(ujJv) (upv)and and{v'v"v"} {v'v"vll) have havebeen beencalculated calculatedby byItzykson Itzykson . and andNauenberg Nauenberg(It-66) (It-66)and andare arelisted listedininTables TablesAl A1and andA2 A2for forpermutation permutationgroups groupsup uptotoS(6) S(6). The TheSU(mn) SU(mn)CG CGcoefficients coefficientscan canbe beexpressed expressedasas [1I1 .. ,B[,,1. WI[l'l~ W 2
C[11'1.8' "'W{I"W~,[lIl/l,BI/"I/W{'I'I/W~'
d"l .. ,B["l'[l'l~ ='"' L..J [v'],8'O"I1',[V Il ]plI«1"#,"
d"19,W 1
O'IW~,O'IIW{'
(5)
dl'l 4> ,W2
""W~,J'"W~'
94>
2.2. The CGcoefficients coefficients of of permutation permutationgroups groups The CG The CG coefficients of the permutation The CG coefficients of the permutation group group S(f) S(f) for for the the Kronecker Kronecker product product [0'] [a]xx [tt] b]isis denoted denotedby by CJ:Jf.',';:m2' where whereml, ml,m2 m2 and andmmcan canbe beunderstood understood either either asasthe the component componentindices indices of r j rj- 1... . . .r2rl r2rl indexed indexedininpage pagedecreasing decreasing ofthe theYamanouchi Yamanouchibases basesororthe theYamanouchi Yamanouchisymbols symbolsrJrJ-l order order(Ha-62). (Ha-62).The TheCG CGcoefficient coefficientcan canbe befactorized factorizedasas(Ha-62, (Ha-62,Ch-Sla) Ch-81a)
~kii:~,,
(6)
where where f3/3 == 1,2, l , 2 ,... . . .(ujJv) (upv) and and p,8 = 1,2, 1,2,. .... .(ujiii) (CjiP) are are multiplicity multiplicity labels, labels, while while cp],P,~ 4m1,prna are the S(f) > S(f 1) ISF and S(f 1) CG coefficient, respectively. Crl~:;"'2 are the S(f) ::> S(f - 1) ISF and S(f - 1) CG coefficient, respectively. By By applying applyingthe thefactorization factorization(6) (6) hf 2 times timeswe weget get
c!~!;F]'
and
(7)
where sum of where the the last last factor factor isis the the S(h) S(f1) CG CG coefficient, coefficient,and and I::;:~~~:!~2I" I ~ ~ ~isis;aaY sum~ of products ~ products ~ , of ,of the ~ the S(f) ::> S(f -1) ISF, S(f 1) ::> S(f 2) ISF, . . . and S(h + 1) ::> S(fd ISF, s ( f ) 3 S(f - 1) ISF, S(f - 1) > S(f - 2) ISF,. . . and S(fl+ 1) > S(f1) ISF, rm,B,II',B'
"ml ,,' ,l'm2l"
= '"' d"J,B.l..iilfi diilfi,~l~ L..J fi~ ...~
"",I'I'
"7?,jJp.
' ••
dvl~Vl,B' ",,' ,1'1"
,
(S)
+
[B], .. .[&I and [0"] [a']are are partition partition labels labels of of the the groups goups S(f S(f --1), I), S(f S(f --2), 2),... ... S(h S(fl + 1)1) where [u], [a],[~], where ... [u] and and S(h), S(fl),respectively. respectively. and new method, method, the theeigenfunction eigenfunctionmethod, method,for forfinding findinggroup grouprepresentations representations emerged emergedfrom fromthe the AAnew (Ch-84b,Ch-S2) Ch-82). There are aretwo two developmentofthe of thenew new approach approachto togroup grouprepresentation representationtheory theory (Ch-S4b, development . There versionsof of the the eigenfunction eigenfunctionmethod method for for calculating calculatingthe the permutation permutation group group CG CG coefficients. coefficients. One One versions the genealogical genealogicalmethod method (Ch-S4a) (Ch-84a) by by which which the the S(f) S(f) ::>>S(f S(f --1)1)ISF ISF and and S(f) S(f) CG CG coefficients coefficients isis the are calculated calculated recursively. recursively. One are One isis the the nongenealogical nongenealogicalmethod method based based on on the the fact fact that that the the CG CG coefficientsof of S(f) S(f) results results from from diagonalizing diagonalizingthe the second secondkind kind of of the the complete completeset set of of commuting commuting coefficients operators (CSCO-II) (CSCO-11) of of S(f) S(f) in in the the Kronecker Kronecker product product space. space. The The CSCO-II CSCO-I1of of S(f) S(f) isis aa suitable suitable operators 2 linear combination combination of of the the ff -- 11two-cycle twecycle class class operators operators and and isis the the analogy analogy of of (J (J2, for the the linear , JJz)z ) for rotation group. group. ItIt isis the the latter latter method method that that isis used used here here to to re-compute re-compute the the S(f) S(f) CG CG coefficient coefficient rotation with as as many many as as possible possible symmetries symmetriesand and consistent consistent overall overall phases phases for for ff equal equal up up to to six six (part (part of of with 1 the results results have have been been reported reported in in Ga-S5 Ga-85l). The overall overall phase phase convention convention isis the ) . The (9)
=
(mlm2) =min min means means first first taking taking the the index index ml ml as as small small as as possible possible and and then then taking taking the the where (mlm2) where index m2 as small as possible (the maximum Yamanouchi symbol r j r j 1 . . . r2rl corresponds to index m2 as small as possible (the maximum Yamanouchi symbol rJrJ-l ... r2rl corresponds to the smallest index m = 1). the smallest index m = 1) .
3
The Algorithm
The CG coefficients of the permutation group have many symmetry properties, which are closely related to the properties of the SU(mn) > SU(m) x SU(n) A CFP.
I
& 1 ( ~ ~ 2= ~ &l(Wlvp) ,3) = s i g n ( c k ~ ; ~ ~(m,ml)=min) &, .
(lob)
The phase factors E ~ ( U ~ Vfor ZU S(2)-S(6) ~) are listed in Table A3.l
where [C] is the conjugate partition of [a],etc., A& is the phase factor associated with the Yamanouchi basis (Ha-62), (mlm2) = max means first taking the index ml as large as possible and then taking the index mz as large as possible. Notice that the CG coefficient for S(2)-S(6) tabulated in Ch-81b do not have a consistent overall phases, but otherwise the results in Ch-8lb and Ga-85 are identical. The phase differences between the two are shown in p. 211 of Ga-85.
here are three printing errors in the tables (Ga-85). The corrections are indicated below: Table 4.1
Table 4.4
4
Tables of the SU(mn)3SU(m)xSU(n) Coeficients of Fractional Parentage
The S(f) > S(f -1) ISF for f = 3-6 have been published in Ch-84a,' which are consistent with the CG coefficients in Ga-85 both in phase and multiplicity separation. Recently, an improved algorithm and code have been developed for computing the S(f) > S(f - 1) ISF recursively which enables us to extend the calculation for higher f (Ch-89b). Schindler and Mirman (Sc-77, Sc-78) calculated the CG coefficients of S(2)-S(6) by using the projection operator method. A serious shortcoming of their results is that the multiplicity separation is entirely arbitrary and thus the CG coefficientsfail to satisfy certain symmetries. Li and Zhang (Li-87) calculated the S(f) 3 S(f - 1) ISF for f = 3-6 by using the double-coset decomposition technique. Vanagas (Va-72) obtained analytic expressions for the S(f) > S(f - 1) ISF for the product [f - 1,1] x [f - 1,1], the only case which is always multiplicity free for any
f. 3. The Let
s
s(f1)
x S(f2) ISF
be the S(f) > S(f1) x S(f2) bases in the x- (- and q = (z,() spaces respectively. The S(f) 3 S(fi) x S(f2) ISF are defined as the coefficients in the following expansion:
/3 = 1,2, . . . (apv),
p1 = 1,2,. . .(atptv'), PIt = 1,2, . . .(aI1p I1I/It ),
where the square bracket indicates the coupling in terms of the CG coefficients of S(fl) and S(f2). Letting f2 = 1 and ignoring the quantum numbers [a"] = p]= [u"] = [I] and the redundant multiplicity labels 0,q5 and T, Eq. (14) can be written as
where
c F J ~ ! $ is~ ~the S(f) 3 S(f - 1) ISF.
4. The calculation formulae for the CFP It can be proved (Ch-8la, Ch-89a) that the SU(mn) precisely the S(f) 3 S(f1) x S(f2) ISF
> SU(m) x SU(n)f2-particle CFP are
---
'~hereare some inconsistencies on the overall phase in the table of Ch-84a. The entries in the following rows or columns should reverse their signs simultaneously, Table 17f, the row headed by [32][311] Table 19d1the column headed by [313] Table 21d1the column headed by [23] Table 22f, the column headed by [214] Table 24e, the row headed by [32][221]
5
The Algorithm
For f 2 = 1, after ignoring the superfluous quantum number 8,4, T, [a"], b"] and [v"], Eq. (16a) becomes i.e. the S(f) > S(f - 1) ISF is the SU(mn) > SU(m) x SU(n) one-body CFP. The S(f) 3 S(fi) x S(f2) ISF, and thus in turn the SU(mn) > SU(m) x SU(n)f2-particle CFP, can be expressedin termsofthe S(N) > S(N-1 ISF, N = 1,2,. . . , f and the S(f) 1S(fl)xS(f2) subduction coefficients (Ch-83b, Ch-87),
(
[ I , Tv'v" m' mu
) [i.. the transformation coefficients
between the S(f) Yamanouchi basis i[v]m) and the S(f) 3 S(fl) x S(f2) basis l[v], [a nonstandard basis of S(f)].
m' m"
Using Eq. (8), (17a) can be written as
When the multiplicity label T is redundant, Eq. (17a) is reduced to
Equation (17) provides a convenient method for calculating the many-particle CFP from the permutation group CG coefficients and the subduction coefficients. It is to be noted that here we do not need the SU(m) and SU(n) Racah coefficients.
5 . T h e properties of the CFP The CFP obey the following unitarity conditions
where 6 p ~ , ~ , ~ B ~ r ~ ~ representation of a, etc.)
=
. . . (notice that here 5 is not the contragredient
~ p ~ B ( , ~ a 6p1pt60,sl l P l ~ ~ ~ a ~ ~ l i f ~ ~ ~
Tables of the SU(mn)>SU(m)xSU(n) Coeficients of Fractional Parentage
6
In the Appendix we shall prove that:
= €1(vlvtlv~ )
€1
(Igb) (19~)
The phase factors (-)' and (-)q(vlvllv*)are listed in Table A4. Notice that the phase factor €l(vlv"v) for multiplicity free cases is factorizable in contrast to the phase factor E ~ ( v ~ uin~ v ) Eq. (10a). The phase factor c l ( y v2v) for multiplicity free cases could be factorized if we abandon the absolute phase convention Eq. (9). However, the phase convention Eq. (9) is crucial for our tables to be valid for both the group chains U(mn) > U(m) x U(n) and SU(mn) > SU(m) x SU(n) (see Sec. 8). From the theory of unitary group representation (Wy-74, Su-80) we have the following symmetries:
where €2 and t3are phase factors, ho(SUm) is the dimension of the irrep [a] of SU(m), etc., [a], b] and [GI are representations contragredient to [a], b] and [v], respectively. Notice that all the phase factors here and in the following, 4, i = 1,2,3.. . and Eil i = 1,. . . 7 , depend on all quantum numbers in the CFP. (iv) From (17b) as well as the symmetries of the subduction coefficients (Ch-83b), and those of the permutation group CG coefficients given in Eqs. (10)-(12), we have
where hv is the dimension of the irrep [v] of the permutation group. (v) Similarly we have
For the identity representation 101 we have [vl,P[ol[Cl (vi) C~vllplol,l,~oI[oI~ol = ~ v v ~ ~ ~ B. I B ~ ~ ~ ~ ~ ~ c c ~
The Algorithm
When [a], [a'] and [a"] are all symmetric or antisymmetric representations, we have (vii) ~
[vlT,Vl[rl+ ~ w ~ l ~ , l [ P ~ l ,=[6vp6v1pi6w11pu6r+ v ~ ~ ~ ~ a ~ ~ P ~ ]
,
C[vl"[l'
l["l*
[ v ~ l [ l ' ~ l [ p i ~ , [ v ~ l= [ l '~vfi' a vl lfi1*vllfi116r+ ~~
1
(viii) From (26) and (22) we have
.
The properties (i), (ii), (iii) and (vi) stem from the unitary group, whiIe the symmetries (iv), (v), (vii) and (viii) stem from the permutation group. It is seen that the interplay of these two groups greatly deepens our understanding of the'symmetries of the CFP. In passing we note that the SU(12) > SU(4) x SU(3) CFP evaluated with much labor by Matveev and Sorba (Ma-78) are trivially found from Eq. (27b) with the phase choice E, = 1,
6. The phase convention From (17) we see that the phase of the CFP is decided by the phases of the CG coefficients and the S(f) 1S(fl) x S(f2) subduction coefficients. The phase for the latter is decided by the convention (Ch-84b, Ch-87)
where m is the index of the Yamanouchi symbol. From (6) and (17) we see that the phases of the CG coefficients, the S(f) > S(f - 1) ISF and the SU(mn) > SU(m) x SU(n) CFP are interrelated. In Ch-84a the following convention is used for the S(f) > S(f -1) ISF: All the partitions are arranged in the order of decreasing row symmetry from top to bottom in the corresponding Young diagrams. For instance, the ordering for the partitions of S(4) is [4], [31], [22], [211] and [14]. The component indices ([a'], [CL']) are ordered similarly, such (PI, Nl), (141, [311),. . . (PI,[l4I),([311, PI), (WI, [31l), . . . ([311, [I4]),. . . ([I4], [I4]). [vlB'[u'lP'} ~ , ~ ,with ~ , (~d , p l ) as the component indices for given The overall phase of the ISF vector { c
.
/3 and [v] = [yv2 . . .vr], UI# 0 and [v'] = [yv2 . .vr - 11 is then determined by demanding that the first nonvanishing component of the vector be real positive. We found that the above phase
8
Tablea of the SU(mn)>SU(m)x SU(n) Coeficients of Fractional Parentage
convention for the S(f) > S(f - 1) ISF and the phase convention Eq. (9) for the permutation group CG coefficients are almost the same for f = 2-6 with the only exception for the ISF's associated with [2212]x [313] + [2212](listed in the last column in Table IV-1-95e and 95f). (Notice that the CG coefficient or ISF for [2212]x [313] + [2212]were not included in Ga-85 and Ch-84b, therefore the assertion that the results in Ga-85a and Ch-84b are consistent remains valid). 7. The evaluation of the CFP With the S(f) > S(f - 1) isoscalar factors in Ch-84a or CG coefficients of the permutation group in Ga-85 and the S(f) > S(fl) 1S(f2) subduction coefficients in Ch-83b or Ch-87 as input (in fact they are recalculated by the program with the special care that the phase and multiplicity separation are identical with those in Ga-85 and Ch-87), the SU(mn) > SU(m) x SU(n) CFP have been calculated for systems with up to six particles. The CG coefficients for [a] x b ] [v] calculated in Ga-85 only involves those b] which are below [a], and [a] is no lower than the self-conjugate partitions. While in calculating the CFP we need all CG coefficients. In order to have a consistent phase and multiplicity separation, the remaining CG coefficients have to be found by using the symmetries of the CG coefficients (10)-(12). can, obtain ~ ~the~ For example, by using (10)-(12) from the CG coefficient c [ ~ ~ $we~ following 23 CG coefficients,
etc. 8. Discussion We see that it is very convenient to use partitions as labels for all unitary groups, so that the universal tables for the U(mn) > U(m) x U(n) CFP can be set up which are suitable for arbitrary m and n. A program package has been written for computing the SU(mn) > SU(m) x SU(n) CFP. The programs are totally automatic and have multiple uses. The only input data are the trivial CG coefficients of S(2). It gives (i) the Young tableaux of S(f ); (ii) the Yamanouchi matrix elements of all the transpositions of S(f); (iii) the CG series (i.e. the multiplicity ( u p ) in Eq. (2)) and CG coefficients of S(f); (iv) the S(f) 1 S(fl) x S(f2) subduction coefficients; (v) the SU(mn) > SU(m) x SU(n) CFP. All the CFP tables have been checked by using the orthonormal requirement for row and column vectors of the table with entries in the form of square root of fractions. Thus it is a check for both the CFP calculation and the correctness of converting decimals to square roots of fractions. So and Strottman (1979) used a similar method to calculate the U(6)> U(3) x U(2) one-body CFP. However, they have not identified the CFP with the S(f) > S(f - 1) ISF, and have to use the Racah coefficientof SU(3) and SU(2) in their calculation. Harvey (Ha-81) and Somers (So-84) calculated some of the S(f) 3 S(f - 1) ISF for f 5 6, which they called the I( coefficient, based on the recursive formula developed by Hmermesh (Ha-62). The phase convention in So-84 is identical to ours. By using the same formula (17b)
~
9
The Algorithm
Somers (So-84) also calculated the SU(mn) > SU(m) x SU(n) three-body CFP (fl = f 2 = 3) for [v]= [I6],[33] and [23]. The SU(4) > SU(2) x SU(2) CFP calculated by Jahn and Wieringen (1951), and the U(6) > U(3) x U(2) CFP calculated by So and Strottman (1979), Somers (1984) and Harvey (1981) are covered by our results. For multiplicity free cases, the result here is identical with Somers' (So84), but differs from So and Strottman (So-79) by phase factors. For the cases with multiplicities, the one-body CFP in Se79, Ha-81 and So-84 do not have the symmetries (22)-(24), while ours have due to the symmetry imposition procedure (Ch-84a), and therefore are much simpler as can be seen from the following comparison: CFP from Table IV-1-28d
CFP from p. 169 of So-79
CFP from Table IV-1-28d
CFP from p. 169 of So-79
-,IC[3211,P[421[3211 [3111[321[221l,[~l[l1[11
=
CFP from Table IV-1-58d
CFP from p. 156 of So-84
It is seen that our results only involve the prime numbers 2, 3 and 5, whereas So and Strottman's results involve the prime numbers as high as 137, and Somers' involves the prime number 103. This shows the importance of the symmetry imposition for the multiplicity separation (Ch-84a, The difference between the results with and without symmetry imposition becomes more dramatic for the many-particle CFP. For example, from our Table IV-5-61e(2) and Somers p. 184 we have the following three particle CFP,
Tables of the S U ( T ~ ~ ) ~ S U ( ~ ) XCoeficienta S U ( ~ ) of Fractional Parentage
where the first and third columns are our results, the four coefficients in the same column belong to (el+)= (1,I), (1,2), (2, I), (2,2). Therefore, without symmetry imposition, the many-particle SU(mn) CFP cannot be expressed as square roots of simple fractions. The symmetry imposition is the most difficult problem in the calculation of the SU(mn) CFP due to the fact that the multiplicities increase rapidly with the particle number f . For S(6), the maximum inner multiplicity is equal 5, while for S(8), it reaches 17 (It-66). If we are content with decimal coefficients rather than the nice form of a square root of simple fractions, we can ignore the symmetry imposition and take what the computer gives us. Perhaps thb is the only possible way of getting the SU(mn) CFP for f > 6. Bickerstaff e t al. (1982) published the 3jm factors for SU(6) > SU(3) x SU(2) which differ from the SU(6) > SU(3) x SU(2) CFP by weight factors and phase factors. Some of their 3jm factors are imaginary. They choose to calculate the 3jm factors for those irreps of SU(3) and SU(6) which are contained in the powers ([I] x [ i ] ) ~=, ~ 1-3. Therefore there is very little overlap between their tables and our tables. As pointed out by Bickerstaff et al. (1982), the phase prescription used by So and Strottman (1979) results in different phases for the CFP involving representations that are equivalent under SU(6) and thus their tables apply only to U(6) not SU(6). This inconsistency in the SU(6) phase is a serious problem, not simply a matter of aesthetics. Our phase prescription is free from this inconsistency, and therefore our tables apply both to U(mn) > U(m) x U(n) and SU(mn) > SU(m) x SU(n). The reason is as follows: the overall phase of the CFP is decided by the orderings of the partitions [v'], [a']and b'], while all these orderings remain unchanged under the restriction of U(N) to SU(N), where N = m, n and mn. For example, according to our convention on the ordering of partitions, if the partition [vl , . . . ,vN] precedes the partition [XI, .. . ,AN], then the partition [vl h, . . . ,v~ h] precedes [A1 h, . . . ,AN h], h being an integer. Now let us take the SU(4) > SU(2) x SU(2) case as an example. Obviously, Tables 11-1-7a and 7b are identical (including the phase) with the following two tables:
+
+
+
+
) U(2) x U(2) to SU(4) > SU(2) x SU(2), the above tables In other words, on restricting ~ ( 4 >
The Algorithm
go over to Tables 11-1-7a and 7b. 9. Appendiz 1. Proof of Eq.(19) As can be seen from Eq. (5) that the symmetry (19) for the SU(n?.n) > SU(m) x SU(n) CFP results from the following symmetries of SU(mn), SU(m) and SU(n) CG coefficients under the interchange of the indices:
Since the Yamanouchi basis of the permutation group S(f) is the special Gel'fand basis of the unitary group SU(f) (Mo-66)) and the S(fl) x S(fi) T S(f) induction coefficient (IDC) or the outer product reduction coefficient of the permutation group is the CG coefficient of unitary group for the special Gel'fand basis (Ch-89a), the phase factor ~l(v'vllvT)is also the phase factor induced from the interchange of indices for the IDC (see Eqs. (4153b) and (7-101) in Ch-89a) and its value depends on the absolute phase convention for the IDC. On the other hand the IDC is also related to the S(f) 1 S(fl) x S(f2) subduction coefficient (see Eq. (4179a) in Ch-89a), and thus the absolute phase convention (28) determines the phase factor (l(vlv'tvT). The phase factors &(vlv"vT) for up to S(6) have been tabulated in Table 5 of Ch-87 and Table A2 of Ch-89a. Later it was proved in Li-89a that the phase factor (l(vlv"vT) can be expressed as
where the phase factor (-)" for [v] = [AIAz . . . A
of S(f) is
while the phase factor (-)q(ylyW"r) takes care of the cases where the outer multiplicity is larger than one. Hecht (He-87, p. 119) introduced another phase factor for the unitary group U(n)
and the phase factor ~l(vlvltvT)is changed accordingly into
It can be shown (Li-89a) that
in spite of (-)" # (-)+("). The advantage of the phase factor (-)" over the phase factor (-)#("I is that the former is independent of the rank of unitary groups and thus is used in this book. The phase factors (-)" and (-)q(ylyl'"r) are listed in Table A4.
12
Tables of t h e SU(mn)>SU(m)xSU(n)
Coeficients of Fractional
Parentage
Table A4 is consistent with Table 5 in Ch-87 and Table A2 in Ch-89a except for ~~([21][11][2~1]). According to Table A4 here and Eq. (A.5), ([21][11][2211) should be equal to +1 instead of -1. The reason for this discrepancy is that although the phase convention for the permutation group IDC in Ch-87 and Ch-89a is consistent with the phase convention Eq. (28) for the subduction coefficients in most cases, inconsistency does occur in rare cases. The absolute phase convention for the IDC in Ch-87 and Ch-89a is (see Eq. (4146) of Ch-89a)
el
where w is the normal order sequence indexed in a definite order (see Eq. (4129) in Ch-89a), and w = min means taking the index w as small as possible. For example, for fl = 3 and f 2 = 2, the ten normal ordered sequences in ascending order are (123, 45),(124, 35), (134, 25), (234, 15), (125, 34), (135, 24), (235, 14), (145, 23), (245, 13), (345, 12). In order to be totally consistent with the subduction coefficient phase convention, Eq. (28), the overall phase convention for the IDC should be
where (w, mz) = min means first taking the index w as small as possible followed by taking the index m2 as small as possible. Similar to Eq. ( 6 ) , the S(f) IDC can be factorized into the products of the S(f) > S(f - 1) [Y]T [v']rf outer product ISF Cvlv~va,,and the S(f - 1) IDC (see Eq. (7-253) in Ch-89a). The phase convention (A. 11) dictates the following phase convention for the outer product ISF,
c ~ ~ ; ! ~ ~ ~ ,
where v' and vi are the partition labels for the irreps of the S(f - 1) and S(fi - I), respectively, and (v', vi) = min means first taking the index v' as small as possible and then taking the index vi as small as possible.
The Algorithm
2 . Multiplicity and phase factor table8 Table A l . The inner multiplicities ( u ~ vof)S(3)-S(6). (a) The group Ss
(b) The group S4 141
1311
(c)
1221
12111
[l4]
The group S6
I51
I411
1321
13111
12all
[16]
[218]
[z21]
[Sll]
[32]
1411 x 1411 1411 x 1321 [41] X 13111 1321 x is21 1321 X [3111 [ S l l l x [311]
The group s6.
*TkMt(right) hudily mateha the top (bottom) heading.
Tables of the SU(mn)>Su(m)xSu(n) Coefficients of Fractional Parentage Table A2. The outer multiplicities
{v'u"u)
of S(3)- S(6).
The group S4
The group S2
[31
t211
The group S5
[S]
[41]
[32]
[311]
[221]
[213]
[15]
[15]
[213]
[221]
[311]
[32]
[41]
[5]
The p u p
s6.
[l3l
[2l21
[zll
1311
1221
1211
1311
1221
[alal
1151
[3l2I
[all]
12131
1321
[213]
1. a i ( v l v a v a ) 2. el(wlvpva)
[213]
[221]
[312]
[2131
[z21] [a21]
131'1
[213]'
[161
= el(v2viva) = 1 , if v l or vz is either totally symmetric or totally antisymmetric.
[312]
12141
121~1
12212]2
121'1
121'1
13l2]2
12211~
[3121
12~11
1221212
1z31
[2212]
[3i3]
131~1
(32112
141'1
p14]
12~1
[31S]
13211
14121
[.la
121~1
1a31 (z21al
13211
1331
pi2]
(421
1317
1321)
1421
1511
1.~1
[2212]1
1421
1511
[vll
[312]
1 2 ~ 1 1 11~1
[312]
12~11
1321
12131
[3121
1411
1411
1321
1~121
1 2 1 ~ 1 12121
111 '
IvlB
[.a1
[vil
-
Table A.3. The triplets (ulvavp) whichhave ~ ~ ( v ~ =v -1. ~ v ~ )
16
Tables of the SU(mn)>SU(m)xSU(n)
Coeficients of Fractional Parentage
Table A4. The phase (-I)['] and
[v] (-)Iv]
(-)q('1"2Vr)
161 [51] 1421 [412] 1331 I3211 [23] [313] [2212] [214] [16] - + -
+
+
+ + +
r=1 r=2 + -
(-)~(1211121113211~)
11. The Use of the Tables
The tables are arranged according to the orderings of f , (fl , f2), [u], b ] , [v'] and [v"] , consecutively. Table IV-3-61e is split into two subtables, e(1) and e(2) for lacking of space. The meaning of the table heading is as follows:
[v'l[v"l
[ol x [a1
where N is the common denominator. All the entries represent the square values of the CFP and a minus sign signifies a negative CFP value. If the same table heading ( [ u ' ] ~ ' ] [ u ~ ' ] [ ~or '~]) [v] occurs more than once, it signifies that some kinds of multiplicities are larger than one. From Tables A1 and A2 one can easily infer what kind of multiplicity occurs in the given circumstance. The integers (I), (2) inside the round brackets on the left indicate the ordering of the multiplicity label PI, or PI1, or 8, or q5; while (11) (12) (21) (22) denote the ordering of the multiplicity labels (O,q5) (for up to S(6), the four multiplicities PI, PI', 8 and q5 will never occur simultaneously except for O and q5 with fl = f2 = 3). The ordering of the multiplicity labels T and P is also indicated in the above table. Notice that all these multiplicity indices are not printed. For example from Table IV-3-61e(2) we can find
IT b[ol@bl+ The CFP C~u,l~,,,,,,[,l,lp,IoIIPII is zero if any of the following six generalized triangle relations
Tables of the SU(mn)3SU(m)xSU(n) Coeflicients of Fractional Parentage
are violated:
A(a1p1 v1) , A(aI1pI1vI1 ), A(apv), A{alatra}, A{plplrp}, A{v'v"v) , where the former three are related to the inner product while the latter three are related to the outer product of the permutation groups. The inner multiplicities (apv) and outer multiplicities {vlv"v} are listed in Tables A1 and A2, respectively (the trivial cases for [v] x [f] = [v] and [v] x [lf] = [C] are not listed in Table Al).
111. Tables
Table I 1. [3] x [3] [21] 2. 3. [13]
1-1 < 312,1> a 25 a-b 25 a 25
4.[21]x[3] 5. [21] 6. [I3]
a-b 25 a-b 25 a-b 25
26 26 26
7.[13]x[3] 8. [21] 9. [13]
a 25 a-b 25 a 25
26 26 26
1-2 < 311,2 > 26 26 26
Table I1 11-1 c 413,l > 11-2< 411,3 > a 27 30 a-b 27 30 a 27 30 a-b 27 30 27 30 a
11-3< 412,2> a 33 a-c 33 a-b 33 a-c 33 a 33
6. [31]x[4] 7. [31] 8. [22] 9. [211] 10. [14]
a-b 27 a-c 27 a-c 27 a-c 27 a-b 27-28
30 30 30 30 31
a-c 33 a-d 33 a-d 33-34 a-d 34 a-c 34
11.[22]~[4] 12. [31] 13. [22] 14. [211] 15. [14]
a a-c a-c a-c a
31 31 31 31 31
a-b a-d a-b a-d a-b
1. 2. 3. 4. 5.
[4] x [4] [31] [22] [211] [14]
28 28 28 28 28
34 34 34 35 35
11-1< 413,l> 11-2< 4(1,3>
11-3C 4(2,2>
16. [211] x 17. 18. 19. 20.
[4] [31] [22] [211] [I4]
a-b 28 a-c 28-29 a-c 29 a-c 29 a-b 29
a-c 35 a-d 35 a-d 35 a-d 35-36 a-c 36
21. [i4] 22. 23. 24. 25.
[41 [311 [221 [2111 111 '
a 29 a-b 29 a-b 29 a-b 29 a 29
31 31 31-32 32 32
a a-c a-b a-c a
36 36 36 36 36
Table I11
111-1< 514,l >
111-3< 513.2 > 111-4< 512,3>
a a-b a-b a-b a-b a-b a
37 37 37 37 37 37 37
55 a-c 55 a-c 55 a-d 55 a-c 55 a-c 55 a 55
69 69 69 69 69 69 69
8. [41]x[5] 9. [41] [32] 10. 11. [311] 12. [221] 13. [213] 14. [I5]
a-b a-d a-d a-e a-d a-d a-b
37 37 38 38 38 38 38-39
46 46 47 47 47 47 47-48
a-c a-e a-f a-f a-f a-e a-c
55-56 56 56 56-57 57 57 57
69-70 70 70 7&71 71 71 71
15. [32]x[5] 16. [41] 17. [32] 18. (3111 19. [221] 20. [213] [I5] 21.
a-b 39 a-d 39 a-e 39 a-e 39 a-e 40 a-d 40 a-b 40
48 48 48 48 49 49 49
a-c a-f a-f a-f a-f a-f a-c
57-58 58 58 58-59 59 59 59-60
72 72 72 72-73 73 73 74
22. [311] x [5] 23. [41]
a-b 40 a-e 40-41
49 49-50
a-d 60 a-f 60
74 74
24. [311] x 25. 26. 27. 28.
a-e a-e a-e a-e a-b
41 41 41 42 42
50 50 50 51 51
a-f a-f a-f a-f a-d
74-75 75-76 76 77 77
a-b 42
51
[32] [311] [221] [213] [I5]
29. [221] x [5]
a
60 61 61-62 62-63 63
The Algorithm
30. 31. 32. 33. 34. 35.
[41] [32] [311] [221] [213] [15]
111-1< 5141 > 111-2< 511.4 > 51 a-d 42 a-e 42-43 51-52 a-e 43 52 52 a-e 43 a-d 43 52 52 a-b 43
36.[213]x 37. 38. 39. 40. 41. 42.
[5] [41] [32] [311] [221] [213] [I5]
a-b a-d a-d a-e a-d a-d a-b
44 44 44 44 44-45 45 45
43. [I5]x [5] 44. [41] 45. [32] 46. [311] 47. [221] [213] 48. 49. [15]
a a-b a-b a-b a-b a-b
45 45 45 45 45 45 45
53 53 53 53 53-54 54 54
111-3< 5(3,2> 111-4< 512,3> a-f 63 77-78 a-f 63-64 78 a-f 64 78 a-f 64 78-79 a-f 65 79 a-c 65 79
65 65 65-66 66 66-67 67 67
79 80 80 80 81 81 81
a 67 a-c 67 a-c 67 a-d 67-68 a-c 68 a-c 68
81 81 82 82 82 82 82
a-c a-e a-f a-f a-f a-e a-c
a
68
Table IV
1. [61 x 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
IV-1< 615,1> a 83 [511 a-b 83 [421 a-b 83 [4111 a-b 83 [331 a 83 [3211 a-c 83 [3l31 a-b 83 83 [23~ a [2212] a-b 83 a-b 83 a 83 [I6]
PI
w41
IV-3< 614,2 > IV-4< 612,4 > a 172 257 a-c 172 257 a-d 172 257 a-d 172 257 a-b 172 257 a-f 172-173 257-258 a-d 173 258 a-b 173 258 a-d 173 258 a-c 173 258 a 173 258
IV-5 < 613,3> a 348 a-c 348 a-d 348 %e 348 a-b 348 a-e 348-349 %e 349 a-b 349 a-d 349 a-c 349 a 349
173 173-174 a-h 174
258 258-259 259
a-c 349 %f 349-350 a-h 350
174-175 175 175-176 176 176-177 177 177-178
259-260 260 260-261 261 261-262 262 262-263
a h 350-351 a-f 351 a-i 351-352 a h 352-353 a-f 353-354 a-h 354 a-f 354-355
12. [51]x [6] 13. [51] 14. [42]
a-b 84 a-d 84 a-e 84
129 129 129
a-c a-f
15. [34] x [411] [33] 16. [321] 17. 18. [313] [23] 19. [2212] 20. [214] 21.
84 a-d 84-85 a-e 85 a-f 85 a-d 85-86 a-e 86 aid 86
129 129-130 130 130 130-131 131 131
a-i a-g a-j a-i a-g a-h a-f
a-f
Tables
IV-1 < 6)5,1> a-b 86
IV-2< 611.5 > 131
IV-3< 6)4,2 > IV-4 < 612,4 > a-c 178 263
IV-5 < 613,3 a-c 355
a-d
a-d
355
a-b a-e a-f a-f a-f a-g a-f a-f a-f a-e a-b
a-h
a-h 355
a-i a-j a-i a-j a-j a-i a-i a-h a-d
a-i a-i a-i a-i a-i a-i a-i a-h a-d
356-357 357-358 358-359 359-361 361-363 363-364 364-365 365-366 366
a-b a-f a-f a-g a-e a-g a-g a-e a-f a-f a-b
a-d a-i a-j a-j a-i a-j a-j a-i a-j a-i a- d
a-e
366
a-i 367-369 a-i 369-370 a-i 370-371 a-i 371-374 a-i 374-375 a-i 375-376 a-i 376-378 a-h 378-379 a-e 379
a 96 a-d 96 a-f 96 a-e 96 a-f 96-97 a-g 97 a-e 97 a-f 97-98 a-f 98 a-d 98 a 98
a-d a-g a-i a-i a-g a-j a-i a-g a-i a-g a-b
a-b 379 a-f 379 a-i 379-380 a-i 380-381 a-i 381-382 a-i 382-383 a-i 384-385 a-i 385-386 a-i 386 a-f 386 a-b 386-387
a-c 98 a-e 99 a-g 99-100 a-g 100-101 a-g 102 a-g 102-104 a-g 104-105 a-g 105-106 a-g 106-107
a-f a-j a-j a-j a-j a-j a-j a-j a-j
a-e a-i a-i a-i a-i a-i a-i a-i a-i
a-h 366-367
387 387-388 388-390 390-393 393-394 394-397 397-399 399400 401-402
>
The Algorilhm
> IV-4 < 612,4 >
IV-5 < 613,3 > a-i 402-403 a-e 404
IV-1 < 615,1> a-e 107-108 a-c 108
IV-3 < 614,2 a-j 221-222 a-f 222-223
a-b 108 a-f 108-109 a-f 109-110 a-g 110 a-e 111 a-g 111-112 a-g 112-113 a-e 113-114 a-f 114 a-f 115 a-b 115
a-d a-i a-j a-j a-i a-j a-j a-i a-j a-i a-d
a-e 404 a-h 405 a-i 406-407 a-i 407-408 a-i 408-409 a-i 41&412 a-i 412413 a-i 414-415 a-i 415-416 a-h 416-417 a-e 417
a a-d a-f a-e a-f a-g a-e a-f a-f a-d a
115 115 115-116 116 116 116-117 117 117 117-118 118 118
a-b a-g a-i a-i a-g a-j a-i a-g a-i a-g a-b
a-b 417 a-f 417 a-i 417-418 a-i 419-420 a-i 420-421 a-i 421-422 a-i 422-423 a-i 423424 a-i 424-425 a-f 425-426 a-b 426
a-b a-e a-f a-f a-f a-g a-f a-f a-f a-e a-b
118 118 118-119 119 119-120 120-121 121-122 122-123 123 123 123
a-d a-h a-i a-j a-i a-j a-j a-i a-i a-h a-d
a-d a-h a-i a-i a-i a-i a-i a-i a-i a-h a-d
a-b a-d a-e a-f a-d a-e a-f a-d
123 123-124 124 124 124 125 125 125
a-c 250 a-f 250 a-h 25&251 a-i 251 a-g 251-252 a-j 252 a-i 252-253 a-g 253
437 a-f 437 a-h 437-438 a-h 438-439 a-f 439-440 a-i 440-441 a-h 441-442 a-f 442
308-310 310
a-c
426 426 426-427 428-429 429-430 43&432 432433 433-434 434-436 436-467 437
108. 109. 110.
[2'12] [214] [I6]
IV-1 < 615,l a-e 125-126 a-d 126 a-b 126
111. [I6] x 112. 113. 114. 115. 116. 117. 118. 119. 120. 121.
[6] [51] [42] [411] [33] [321] [313] [2'] [2'12] [214] [I6]
126 126 126 126 126 a-c 126-127 a-b 127 a 127 a-b 127 a-b 127 a 127 a
a-b a-b a-b a
> IV-2< 6(1,5 > 170 170 170 171 171 171 171 171 171 171 171 171 171 171
IV-3 < 614.2 a-h 253-254 a-f 254 a-c 254 a a-c a-d a-d a-b a-f a-d a-b a-d a-c a
254 254 255 255 255 255 255 256 256 256 256
> IV-4< 612,4 > 345 345 346
IV-5< 613.3 > a-h a-f a-c
442-443 443 443
a a-c a-d a-e a-b a-e a-e a-b a-d a-c
443 443 443-444 444 444 444 444-445 445 445 445 445
a
Table 1-1 <3 1 2,1> la.[ 3 lX[ 3 I
Table 1-2 01 l , b
26
Table 1-2 <3 1 1,2> la.[ 3 ]x[ 3 ]
6b.
Table 11-1 <4 1 3 . b
Table 11-1 <4 1 3 . b I..[
4 ]x[ 4 ]
7 . 4 31 ]x[ 31
I
Table 11-1 c4 13,1>
13b.
Table 11-1 c4 13.b
m.
Table 11-2 <4 1 1.32
30
Table 11-2 <4 1 1,3> la.[ 4 ]x[ 4 ]
71.[ 31 ]x[ 31 ]
Table 11-2 <4 1 1.3>
14b.
Table II-2 c4 1 1,3>
2%.
Table 11-3 c4 122>
Table 11-3 <4 1 2,2> la.[ 4 ]x[ 4 ]
5a.[ 4 ]x[llll]
Table 11-3 41 2,B
10c.
1lb.
Table II-3 4 1
17a.[ 211]x[ 31
I
Table II-3 c4 1
2%.
23l.[llll]x[ 22 1
Table JJI-1 414,1>
Table III-1 <5 14,1> la.[ 5 ]x[ 5 ]
C.[ 5 ]x[21111
Table JJI-1 6 1 4 . 1 ~
llb.
lld.
Table III-1 6 14.b
25c.
Table IU-1 6 14,1>
3 h . [ 221 ]x[ 41 ]
Zlc.
Zla
Table Ill-1 d 14.b
3%.
Table IlI-1 414.1s
38d.
Table III-1 414.b
44b.
Table LU-2 d 1 l,4>
46
Table 111-2 <5 1 1,4> la.[ 5 ]x[ 5 I
6a.[ 5 ]x[ 21111
Table IJI-2 6 11.4>
I%.[
llb.
llc.
Ild.
41 ]x[ 221 1
Table III-2 d I l,4>
17c.
Table DI-2 6 1 1.4>
20d.
50 We.
Table Dl-2 <5 1 1,4>
25c.
Table lU-2 d I l,4>
30.4 221 ]x[ 41 1
Zlc.
Table IU-2 41 l,4>
3%.
Table III-2 6 1 1.4s
38d.
Table IU-2 d 11.41
44b.
Table III-3 6 1 3 P
Table III-3 <5 1 3,2> la.[ 5 ]x[ 5 ]
40.
llb.
1lc.
lld.
llf.
1%.
Table IlI-3 4 13>
1%.
Table IU-3 6 1 3 a
19f.
Table IU-3 6 1 3 P
Table III-3 -51 3 b
Sf.
Table m-3 4 1 3 p
Zlc.
Table m-3 6 1 3 . B
3%.
Table m-3 6 13>
3%.
Table IU-3 6 13p
39d.
Table ID-3 6 13,D
43a.[11111]x[ 5
1
Table IU-4 6 123>
Table 111-4 <5 1 2,3> la.[ 5 ]a[ 5 ]
4.3.
Table DI-4 6 12.3s
lob.
1of.
llb.
Table IU-4 6 1 2,3>
lld.
1Zf.
13. llf.
Table IU-4 6 123,
17a.[ 32]x[ 32 ]
Table m-4 6 125
m.
Table llI-4 6 12,P
Table ID-4 6 123>
Table IU-4 6 12,3>
2&.
Table m4 6 12.3,
3%.[ 221 ]x[ 311 ]
Table III-4 6 12,3>
34.
Table III-4 6 123>
38c.
Table DI-4 6 123>
41c.
Table IU-4 41 23>
48a.[11111]x[ 21111
Table N-1 <6 15.b
Table IV-1 c6 1 5.12
14d
Table IV-1 4 15.1>
Ma.[ 51 ]x[3111 ]
Table N-1 c6 15,1>
21d.
mb.
I
41 1 [I1 [Sl ]
[421
[ 41 It 311 1,[11[11 [ 32 11 41 1.[11[11
121111 111
t4111
30
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-1
14111
Table N-1 c6 15,1>
Table N-1 c6 15.1s
Be.
Table N-l e6 15.1>
31b.
[ 221 1 [I]
[3211
[ 411 ]x[ 411 1 [3211 [ u 2 ]
[Pll]
10 0
-2
Table N - 1 c6 15,1>
39c.
Table IV-1 c6 15,1>
Table N - 1 <6 15,1>
4%.
Table IV-1 c6 15.1>
476
Table IV-1 c615.1>
5Oc.
Table N-1 415.1>
53e.
Table IV-1 <6 15.1>
Table IV-1 c6 15.1s
Table IV-1 <6 15.1>
59b.
Table IV-1 <6 15.1>
Table N-1 <615,l>
61b.
Table N-1 c6 15.1>
Table lV-1 <6 15,1>
6 k
Table IV-1 415.1>
Table lV-1 <6 15,lr
6 4 ~
Table IV-1 c6 15.1>
Table N-1 <6 15,1>
Table IV-1 c6 15,1>
Table IV-1 <6 15,1>
Table IV-1 <615,1>
7%.
Table IV-1 <6 l5.1>
ma.[ 222 ]x[ 51
I
Table IV-1 <6 15.1>
826
Table IV-1 c6 15.b
8%.
Table IV-1 <6 15,1>
90c.
Table IV-1 e6 15.1s
Table N-1 c6 15.1>
Table N-1 e6 15.1>
94d
Table IV-1 <6 15,lr
Table IV-1 <6 1 5,1>
98b.
Table N-1 c6 15.b
1066
Table IV-1 <6 15,1>
Table IV-1 <6 1 5.1>
128
Table IV-2 <6 1 1,5>
1Ob.
Table IV-2 <6 1 1,s
Table IV-2 <6 1 1.52
Ma.[ 51 ]x[ 3111 I
(11 [ 21111
[ 51 ]x[3111 ]
[3111 ] [11[11.[ 41
[Zll ]
[21111]
I[ 311 I -8
0
Table N-2 e6 1 1,5>
Zb.
Zc.
Ze.
Table IV-2 c6 1 1.5,
111 1 5
I
[11[11.[ 32 I[ 32 I
nc.
42 IX[ 33 I
Table IV-2 <6 1 1.5>
ne.
[ll [ 32 I [ 72001 [11[11.[ 41 I[ 32 I [1][1],[ 41 I[ 311 1 [11[11.[ 41 I[ 221 1 [11[11.[ 32 I[ 32 I [ll[llv[ 32 I[ 311 1 [11[11.[ 32 I[ D l 1
[ 42 ]x[ 321 1
[ 42
1
225 2430 225 -720 0 3600
[ 42
202.5 -750 2025 0 2400 0
1
[ 33
1
1250 -1500 450 loo0 -1200 1800
[ 321 1
[ 321 ]
[ 321 ]
0 1536 0 5184 480 0
-3200 -960 0 40 lux) 1800
-500 -24 4500 256 -1920 0
Table IV-2 <6 1 1,5>
28e.
Table lV-2 <6 1 1.5>
Table IV-2 <6 I 1 . 5
326
Table N-2 c6 11.5,
36f.
Table N-2 e6 1 1.5
Table IV-2 <6 I 1 . 5
Table IV-2 c6 1 1 . 5
44b.
Table N-2c6 1 1 . 5
49b.
Table N-2 41 1,5>
5%.
Table IV-2 c6 1 1 . 8
57a.[ 321 ]x[ 51 ]
[I] [ 221
1 [321]
[11[11.[ 311
I[
41
[ 321 ]x[ 51 ] [a [321]
][ P l l ]
1 48 192
144 -45
[ll [ 32 I ( 7200) [11[1IJ 32 I[ 41 I [11[11,[ 32 I[ 32 1 [lI[lIs[311 I[ 41 I [lI[lIs[311 I[ 32 I [11[11.[221 I[ 41 1 [11[11.[ 221 I[ 32 I
[ 321 ]x[ 42 [ 42 ]
[ 42 ]
225 -720 2430 0 225
2025 0 -750 2400 2025
3600
0
1
[ 33
1
1250 lo00 -1500 -1200 450 1800
[ 321 1
[ 321 1
[ 321 1
0 5184 1536 480 0 0
-3200 40 -960 lux) 0 1800
-500 256 -24 -1920 4500 0
Table N-2 c6 1 1.5s
Table IV-2 c6 1 1 . 8
bob.
Table IV-2 c6 1 1.5>
61b.
Table IV-2 e6 1 1 . 5
Table N-2 c6 11.5
Table IV-2 <6 1 1.5>
Table IV-2 e6 1 1,5>
64e
111 121111
[ 3111 ]x[ 51
[3111] [11[11.[21111[ 5
1
[Pll]
[21111]
-8
0
1
Table N-2 <6 1 l.5>
Table IV-2 <6 1 1 . 5
Table N-2 c6 1 1,5>
72d
Table IV-2' c6 1 1 . 5
Table IV-2 c6 1 1.5>
76.[3111 ]x[211111
Table N-2 <6 1 1 . 5
80f.
Table IV-2 <6 1 1,5>
Table N-2 c6 1 1 . 5
Table N-2 411.5>
91f.
Table IV-2 <6 1 1.5
Table N - 2 c6 1 1.5
Table N-2c6 1 1.52
%f.
Table IV-2 <611.b
101b.
Table IV-2 <6 1 1.52
105a.[ 2llll]x[ 321 ]
111 [ 21111 4 )
[11[11.[ 21111[ 32 I [1][1],[ 2111][ 311 1
1Mb. 1Wc.
[ 2 l l l l ] x [ 321 ] [3111] 3 -1
[Zzll] I 3
Table IV-2 <6 1 1 . 5
10%~.
Table IV-2 <6 1 1,5>
116c.
172
Table IV-3 c6 1 4,2>
Table N-3 c6 14,B
9d.
lob.
Table IV-3 c6 14.5
14d
Table IV-3 c6 l4.B
180.
Table IV-3 c6 14,>
206
rnb.
Table N-3 c6 14.B
23d.
Table IV-3 <6 14.B
Table IV-3 c6 14.B
Table N-3 <6 14.B
Table IV-3 c6 14,b
Table N-3 e6 14.B
Table IV-3 <614,D
29d
Table lV-3 c6 14.B
Table lV-3 <6 14.5
Table IV-3 <614,b
Table N - 3 414.b
3&.[ 411 1x1 6 1
Table N-3 <614,B
Table IV-3 <6 l 4 , b
Table IV-3 <614,D
Table N-3 e6 14.D
Table IV-3 <6 14.B
Table N - 3 c6 14.b
39c.
Table IV-3 <6 1 4 3
Table N-3 c6 14.B
40c.
Table N-3 c6 14.D
Table N-3 c6 14.B
Table IV-3 <6 14.B
Table IV-3 c6 14,B
46c
Table IV-3 c6 14.B
48f.
Table N-3 e6 14,b
5Zc
Table IV-3 c6 14.B
Mg.
Table IV-3 c6 1 4.B
[ 31 1 [ 21
21600) [ 31 I[ 4 I,[21[21 [ 31 I[ 31 I.[ 21[ 21 [ 31 I[ 31 ].[11][11] [ 31 I[ 22 I,[21[ 21 [ 22][ 31 I,[2][ 21 [ 22][ 31 ],[11][11] [ 2111[ 31 1.1 2I[ 21 [2111[31].[111[111 [ 21 1][ 22 I,[2][ 21
[ 321 ]x[ 42 ]
[Sl] -4000 -300 450 -50 -1800 -1500 4500 6750 2250
[42]
[42]
noo
-noo
-3240 1215 -135 2160 4050 0 2025 6075
0 -3375 -3375 0 4050 -5400 2025 -675
[411] 1000 7500 0 -800 -1800 6000 4500
[411]
[33]
-zoo
mn
750 4500
-750 4500 -2000 4500 -600 450 2700 -3600
2000
4500 600 450
o
noo
0
-3600
[321] 5760 1728 4608 72 2592 -2160 0 4320 -360
[321] 400 5880 2880 -3920 720 -%KO
-1800
o 3600
[321] 40 1452 72 9248 -3528 240 4500 1080 1440
Table W-3 <6 14.D
58c.
Table N-3 <6 14.b
Table N-3 <6 1 4 2
Table IV-3 c6 14.B
[ 321 b[ 321 1
131 1 I111 76800 1 [ 31 I[ 31 I.[ 21[111
42
1
-9216 -9216 I31 I[ 22 I,[ 211111 -4320 131 I[ 22 l.I111[ 21 -864 [ 31 I[ 2111,I 21I111 -8640 -960 131 I[ 2111.I1111 21 -864 122 11 31 1,. 211111 I22 11 31 1.[111[ 21 4320 [ 2211 2111~ 21[111 4320 [ ~ I [ 2 1 l ] . l l l ] [ 2 1 864 -960 [ 21111 31 I.[ 21[111 [ 21111 31 1.[111[ 21 -8640 864 [ 21111 22 1.1 211111 4320 [21llf 22 ].Ill][ 21 [ 2111[ 2111.[ 21[111 9216 [2111[ 211].[111[ 21 9216
I 31 11 31 I,[III[ 21
[ 42
I
[ 42 ] [ 411 1 [ 411 ] [ 411 ] [ 411 ] [ 321 ] [ 321 ] [ 321 ] [ 321 ] [ 321 1 [ 3111 ] [ 3111 ] [ 3111 ] [ 3111 ]
5760 144 6000 m -225 5760 144 4800 480 -7200 750 -8640 -7776 - 4 ~ a -7350 0 -2160 400 -3375 0 19440 -3600 16375 -8640 -7776 -4OOLl 7350 4800 480 -7UY) -750 4800 -480 7201 -750 -8640 7776 4WO 7350 0 19440 -3600 -1895 0 -2160 400 3375 -8640 77-16 4wo -7350 -4800 -480 7UY) 750 5760 -144 -6000 -W W 5760 -144 -6000
9600 -9600 8000 1600 0 0 -1600 -8000
9375 3840 o o IISZO 19x1 -9375 ss4o o o i i s m IWO -2450 0 -12800 -3UY) 0 14400 -WO 2560 o 16000 7680 -2880 -3025 -25600 WKI 0 0 0 6400 0 -7225 0 0 0 WO 2560 0 -16000 7680 -2880 %SO 0 12800 3200 0 14400 %SO o 12800 -3200 o 14400
1600 0 0 -1600 -8000 9600 -9600
7.250 -2330 72u 0 3025 -25600
.mo -2450 -9375 9375
0
16000 7680 0 0 44M) 0 0 -ma o -16000 7680 0 -1uWO 3200 0 -3840 0 0 11520 -3840 0 0 11520 4400
-2880 0 0
-mo 14400 1920 1920
-9600 9600 8000 1600
-m
o
-7350 7350
m
o
-2700 ism
-7m 4000
-8100
0 0 -1600 -8000 8000
-25350 150 -1500 2700
1600 14400
1600 0 0 -1600 -8000 -9600 9600
-MO
4M0
250 4250 8100
nm
-7m 7 m
MO MO
-1500 -150 25350 15m -2700
XOO
14400 1600
8100 6250 -250 -8100
ZYJ
7 m 0 0
-MO
7350 -7350
Table N-3 c6 14.b
Table IV-3 <6 14.b
Table IV-3 c6 14.b
Table N-3 e6 14.b
Table N - 3 c6 14.B
63h
Table IV-3 c6 14.2
Table N-3 c6 14.D
[ 2111 [ 21 [ 60) [ 2111[ 4 1.1 21[ 21 [ 21111 4 IS[11I[11I [ 2111[ 31 I.[ 21[ 21 [11111[ 31 I.[ 21[ 21
[411]
20 20 0 20
[3111 ]x[ 51 ] [ 3 2 1 ] [3111] -8 -5 -8 u) 12 30 32 -5
[Zll]
n -12 18 -3
Table N-3 <6 14.B
Table N-3 <614,s
Table IV-3 c6 1 4 3
Table IV-3 <6 14.D
Table N-3 <6 14.b
Table N-3 c6 14.b
72f.
Table N-3 e6 14.B
Table IV-3 <6 14.5
Table IV-3 <6 14.B
Table N-3 <6 14.5
756
Table IV-3 <6 14.5
76i
Table IV-3 <6 l 4 . b
80f.
Table N-3 c6 1 4 , s
82r
Blh
Table N - 3 e6 14,s
846
[ l l l l ] [ll]
1 222 lx[ 321 I 1 [ 211111
[ 2211 [ 2111[ 31 1.[111[ 21
Table IV-3 <6 14.B
86c.
Table IV-3 <6 14,b
90b.
Table IV-3 <614,s
Table N-3 c6 14,b
Table N - 3 <6 1 4 3
[ l l l l ] [ll]
[2211 ]x[ 411 1 [2211 ] [21111]
[ Z l l ]x[ 33 ] [ 2111[ 31 I.[ 21[ 21 [ 2111[ 22 1.[111[111
Table IV-3 <614,b
Table IV-3 <6 14.B
Table IV-3 <6 14.B
Wh.
Wj.
Table N-3 c6 14,s
Table IV-3 c6 14.B
Table N-3 c6 14.b
[llll] [11] [11111[ 4 1,[111[ 21
[21111]x[ 6 ] [ 211111
Table N-3 c6 14.b
lmd.
[ 2111 [ 21
[21111lx[ 33 [ 321
[11111[31 I,[211 21
1
I
[P l l1
Table N-3 Q 14.-
1m.
Table IV-3 <6 14.D
109d.
Table IV-3 c6 14.5
116a.[llllll]x[ 321
1
Table W-3 <6 14,s
Table IV-4 c6 1 2,4>
Table IV-4 c6 1 54>
9d.
Table N-4 c6 124>
14.
Table N-4 <6 124s
161
Table N-4 c6 1242
I8c.
Table IV-4 <6 124>
20d.
Table N-4<6 124>
1-
[11] [ l l l l ]
[ 51 ]x[llllll]
[21111]
t 2l[lll.[
4 1[11111
-1
Table IV-4 <6 1 2 . b
Table IV-4 e6 1 2,4>
Table IV-4 <6 12,4>
268
nc.
Table TV-4 <6 124
28b.
Table IV-4 <6 124>
[11][llll]
[ 42 ]x[ 321 ]
(22111
[111[21J 31 I[ 2111
Bb.
[Dl11
[21111]
Table N-4 <6 124>
Table IV-4 c6 124s
Table IV-4 <6 124s
Table N-4 e6 124>
Table N-4 <6 154>
Table IV-4 <61 2,4>
Table IV-4 c6 1 24>
Table N4 e6 1 2,4>
Table IV-4 <6 l2,4>
Table IV-4 <6 124>
Table IV-4 <6 1 2,4>
Table IV-4 c6 1 24>
Table N-4 <6 124,
Table N-4 c6 1 24>
Table IV-4 c6 124>
46a.t 33 ]x[ 51 ]
Table IV-4 <6 I 24>
Table IV-4 c6 1 2,4>
Table N-4 <6 124>
Table IV-4 <6 124>
Table IV-4 <6 1 24>
Table N-4 c6 124>
53i
Sf.
1111 122 1 [ 33
I
[ 321 la[ 51 1 [ 321 I [ 321 I
11111 21.1 31 I1 31 1 11111 21.1 22 11 4 1 12
72
[P l l1
Table N-4 <61 2,4>
Table N-4 <6 1 2,4>
Table IV4 c6 1 2,4>
Table IV4 c6 12,4>
596.
Table IV-4 c6 1 24>
Table N-4 c6 12.b
60i
121 [ 31 1
I -1 [21121.[ 31 I[ 31 I [2112).[311122] I211 21.1 31 11 2111 121[ 21.l2.2 11 31 1 21[ 21.t 22 I[ u l l 1211 21,12111[ 31 I 21[ 21.1 2111[ 22 1 1 ZI[21.1 2111~211 [111[111.131I[ 31 1 [111[111.[31 I[ 22 1 I 11 1 I 22 I 1 1 [1111111.[22 11 2111 1 1 1 1 1 3 11 [111[111.[ 21111 22 1 [111[111.[ a l l [ 2111
[ 321 lx[ 321 1
~511[51][421[42][42]~411][411][411][411][33][33][321][321][321][32~][32~]
aMS 150 -375 150 7 -375 7%
w w 750 375 750 150 375 150
aM5
150 -900 YO -900
432 648 0
m
360 0 360 -2352 -2.52
YO 500
iao -1350 -500 YO -SIN
900 750
900 -150
618
160
o 360 648 0 648 432
-1470 -363 180 -882 450 -1125 180 -882 900 90 -450 -1125 9u) 90
-1500 0 100 0
no .as -no -as
i m i m 8M -100 800 0 -100 0 -1500
-900 450 -900 -180 -450 -180 1470
90 1125 90 -2 117.3 982 -263
soo 100 (100
0 0 150 900 1% -1WO -150 -900 -750 -500 -la lM0 750 500
o
o
0 -2250 100
1500 0 -100
pso
o soo
-50 -100 50
o
o
o o
-750 -im
500 -1m
-50 -100
750 150 1500 -150 0
-500 -900 10W 900 0
50 -2250 100 2250 0
-100
soo -ism -800
lm
0
lm
as0 -720
600 .im
-720
xm
ua
8133
-600 8 0 - 3 ~ 400
o
~ 8
o
0 0 800 0 l a -800
o
1440
a 0
0 -800 0
sao
.3M)
800
.m~) -1600 0 o o
~ 8 0 0 - 3 ~ ) 400 1600
600
su)
400
0 -1600 400
-800
300
0 -100 0 -1500
-1500 -600 1500 0
400 -720 400 -720 480
- l a 0 800 0 0
1600 0 800 0 0
-100
0
-1600
960
-360
0 0 0
-1600
1440 -1440 0 0 0
-966 0
.m -1440
.ZOO
240 .=0
200
-1600 200
160 -1600 360 .y~)
Table IV-4 <6 1 24>
121 [ a l l
1
76300 )
[ zl[ 21.[ 31 11 31 1
[ 321 ]x[ 321 1
[ 411 1 [ 411 1 [ 411 ] [ 411 ] [ 3U ] [ 321 ] [ 321 1 [ 321 ] [ 321 ] [3111 1 [ 3111 ] [ 3111 ] 13111 1 [2U1
0 8000
I211 21d 31 11 Z I 9600 [21[ 21.1 31 11 2111 a000 [ 211 21,1zz I[ 31 1 -1600 [ 2lI Zl.[Z? I[ 2111 9600 I211 21.[ 2111[ 31 1 -1600 I21[ 21.1 2111I Y 1 0 [ 211 2l.I 2111[ 2111 0 [111[111,[ 31 11 31 I ~111[111.[31 I[ 22 1 -1600 9600 ~111l111,[31 I[ 2111 ~111[111.~Y1~311-1600 8000 1111[111.[ 22 11 2111 9600 [111[111.[ 2111[31 I 8000 [1111111.I Wilt 221 0 [111~111.1211112111
25350 -7.700 22M
na, -15a -22M
1500 150 150 -1500 22M lSOO -2700 -2233
27M 25350
1600 250 0 0 0 6400 25600 4 0 3375 7200 500 750 -3200 12800 0 14400 -7200 0 0 -7350 -225 0 0 -3840 11520 -1920 -aooO -7m -500 -3200 -12800 -750 o o 17200 -4000 -8100 -16000 0 2560 -7680 -2880 &CHI 7350 0 7350 0 3840 11520 -1920 6000 225 0 4000 8100 -16000 0 -2560 -7680 -2880 4000 -7350 14400 6250 0 -6400 0 0 3600 18375 0 -14400 6250 0 -6400 0 0 -3600 18375 0 4000 -8100 -16000 0 -2560 7680 2880 4000 7350 0 -7350 0 0 -225 3840 -11520 1920 6000 -4000 8100 -16000 0 2560 7680 2880 -4aK) -7350 .7200 500 -3200 12800 0 750 0 -14400 7200 0 7350 0 0 -3840 -11520 1920 -Sax, 225 77D -500 -3200 -128W -750 0 0 -14400 -7aDO -1600 250 0 6UX) -25600 400 0 0 3375
0 -8000 9600 -8000 1600 9600 1600 0 0 1600 9600 1600 -8000 9600 -8OOO 0
-3025 WO 9375
-8640 4320 -9216 -WO 4320 -2250 864 -9375 9216 2250 -864 7225 -960 960 7225 -2250 -864 9375 9216 2250 864 W O -4320 -9375 -9216 -2350 4320 -3025 8640
1
12211 1 [ 2 x 1 1 0
4800 5760
2160 480 -144
4800
480
8640 5760 8640
-7776 144 7776 -19440 19440 7776 144 -m6
0
0 -8640 -5760 -8640 4800 -5760 4800 0
480 -144 *80
-21a
Table IV-4 c6 124>
1111 1 all
1
-1
Iarlil.1 31 11 31 I [211111.I 31 I[211
I211111~31 I[ 2111 I211111J P I [ 31 1
[ 321 M 321 1 1321][3211~3211l3~)I321]I3111][3111][3111][3111][2a][222][2211][~1][2211][21111][~111]
-8m
4m
nm
o
.im
-1m
0
0
-720 480
.960
-W
0
.la0 0
aoo
ism
-a
-8m
-m aoo
o -1sm -800
o
o
1600 0
-1600 o
[211111.I 22 11 all o [ 2Ipll.l a111 31 I [211111.[ 21111 221 -1600 1211111.1~1112111 sm 800 11111~I,I 31 11 31 I 1600 [ l l l l 21,I 31 11 2 1 o 1111121.t 31 11a11 o II[ 21.1 P 11 31 1 -1600 [~~II~I,IPII~II 0 1 11 o IlllI21,I all[ u] -800 IIIII21.1 a111 2111
o 1600 800
400 -tu,
-9m
o
o
-750
5M
-1m
1500
-im
-ism
iom iom
-1m im
0 -1440 .960
-ZW
8m 1500
-7%
-5m
o la
o 9m
2w,
sm
are
480 720 - 4 ~ o -480 -1440 720 -960
-360
-1500 0
7x1 0 -1%
m
im
.im
Ism
o
-m .ZW 7AO
o urn
0 -900 -1om
0 300
432 -360
.180 1470 -900
.ism -ism o
-648
-180
nn
.no
o o
uo
z z ~
-ZW
m
o -648
o
-1m 1600
o
800
-300
-im im
M)O M)O
50
800
3m
o
o ism
-so
-ism o urn
0
1500
-m -im
uo
,100
ism
-1m 0 1500 800
-50
o o
a00 -uK)
o em
1%
im -zzM
o 50
-am
-1600 0
-1om 900 o -500
40 -1440 400 0
1600
o o
-960
-1500 -150 o 7%
-360
360 -as2
w
-900
-11z .882 .363 90 -882 -243
90
-11s 40 -11u -900 -90 n o 243 -180 882 .~OO
.w
w
1470 -180
o
u o
363 882 -11u
-3m 0
360 432
o -Ism im 600
-37s 1% 7% 1%
us 7% -375 -375 -7%
.w -1% -750 -20s -1% -375
ZM -900 1% 500 -9m 1350 500 250
-w, 500 1350 -900 500 1 s -900 -250
Table IV-4 c6 124s
[ 321 1x1 3111 ]
[ll][llll]
[ 21[111,[2111[ 31
[Dl1]
[ 2211 1
[21111]
4
10
-2
I
[111[21,[31 I[ 2111
Table IV4 <6 12,4>
Table N-4 c6 1 24>
Table IV-4 <6 1 2,4>
Table IV-4 <6124>
Table IV-4 <6 124
Table IV-4 c6 12.b
Table IV-4 <6 1 54,
Table IV-4 c6 1 2 b
Table IV-4 <6 12,4>
'TableN-4 c6 124,
Table lV4 <6 I2.b
Table IV-4 c6 1 24>
Table IV-4 <6 1 242
Table IV-4 <6 I 2,4>
Table IV-4 <6 1 2,4>
756.
Table N-4 <6 124>
Table N-4 c6 12,&
79g.
Table IV-4 <6 1 24>
818.
Table N-4 c6 124>
836.
Table IV-4 c6 124>
85d.
Table IV-4 <6 124>
Table IV-4 41 24>
Table N-4 <6 124>
91c.
Table IV4 <6 1 24>
Table N-4 c6 124>
Table N-4 <6 12b
Wb.
Table IV-4 <6 1 2,4>
94d.
Table N4 <6 12,4>
[21[ 41 [ 21[ 21,[2111[ 2111
[111[111,[2111[ 2111
[Dll ]x[3111 ]
Table IV-4 c6 1 24>
Table IV-4 c6 1 24>
Table N-4 e6 1 24>
97b.
Wf.
Table IV-4 c6 1 2,4>
98h
Table IV-4 <6 1 24>
Table IV-4 <6 1 24>
Table lV4 c6 12,4>
Table N-4 <6 1 24>
Table IV-4 <6 1 2,4>
Table IV-4 c6 I 24s
108f.
Table N-4 c6 1242
113d
Table IV-4 <6 1 24>
119b.
348
Table IV-5 c6 1 3,3> la.[
6 ]x[
6 ]
Table IV-5 c6 13.3s
9c.
Table lV-5 c6 1 3 3
16d
Table N-5 c6 1 3.3>
Table N-5 c6 1 3 3
Table N-5 <6 13.3>
Table N - 5 <6 13,s
2 4 ~ [42 ]a[ 51 1
Table N-5 c6 13.P
Table IV-5 c6 13.3
Table N-5 c6 1 3 3
Table N-5 c6 1 3,s
1 21111111 ,
(
5 )
[ 3I [211.[ 2111 211 [ 211[ 211.[ 211[ 211
[ 42 IX[33 I [321] [Zll]
-1
4
4
-1
Table N-5 <6 1 3.32
I211 [ 211 1-1 [ 3 I[211,[ 3 I[ 211 [ 3 I[211,[ 3 I[ 211
[ 3 I[211.[21][ 3 1 13 11 211.I 211[ 211 [ 3 I[ 211.[ 2l][ 211 [ I t 211.t 211t1111 [ 2111 3 I,[ 3 11 211 [ 2111 3 I,[ 21][ 211 12111211.1 3 I[211 I211I 211.[3 I[ 211 12111211.1 211[ 3 1 I21II 211.1 211[211 [ 211[ 211s[ 211t 211 211[ 211B[ 211[1111 1211I1111.I 3 11 211 I211I1111.I ZlI 211
I 42 I42 1 18100 0 45 -45 2073
I 4 2 1 1 4111 [ 4111 [ 33 1 1 3211 I 3 2 1 1 [ 3211 -1800 0 0 0 640 -1600 -2560 0 0 m -m 0 0 0 -1125 1800 -500 250 64 -1440 576 117.5 200 500 -230 -2304 -640 -1936 22.5 1 m 1100 54 1280 0 -7u) 22.5 2500 450 -320 800 1280 -250 64 -1440 576 -117.5 -1800 500 -720 0 XX) 0 -1Mkl 64 -1440 576 45 -1125 XX) 5100 -2250 2324 640 1936 207.5 225 -1000 -100 -M 1280 0 -720 720 0 XX) 0 -1m -64 1440 -576 2880 0 0 0 0 -3136 -2560 10U O 0 a 0 8100 0 0 0 0 3600 lm -1aoo -18100 l a 0 -800 3u)
-= -225
lm
urn
3m
-1m
laoo
0
454 18100
320 1280
-800 -800
-1280 3u)
)XI 321 1
I 3211 0 -640 -320 1280
[ 3211 [ 3211 [3111] [31111 ( 2 2 2 1 [ 2 1 ] [ W l ]
0
0
-lW
urn
-Ern 0
6(
1440 800 4100 0 -640 -8100 0 -360 1440
1280 -720 -1936 576 -1280 -320 -720 19% -320 0 -10% 576
61 4
1440 -1440
n6 .y,6
-2304 64
320 -1280
lm 23W
-1280 0 3136
640
0 0
0 0 0 -1800 8l(, 60 u n ~ ps -1~) 50 M '125 -Mo - W O 4 117.5 500 250 45 -1125 -2yx) a 0 1600 -1800 0 -36~) 1 ~ ) -50 -w - - W O 411 -11= -1600 ~ 1800 0 -3600 0 goo 0 -2880
4000
uTX)
0
lm -m 1 m XX)
1800 1000
lm -1000 XX)
-
1 0 -XX)
.1800
-m
0 1 m - 7 2 0 -MO -30 45
lm
no
1 -l75 .
0
Table 1 ' - 5 c6 1 3 . 3
28f.
Table lV-5 c6 1 3 . P
Table IV-5 <6 13,3>
Mb.
Table IV-5 <6 1 3 . 3
Table N-5 c6 13,*
31f.
Table N-5 c6 1 3.3>
Table IV-5 <6 13.3s
[Ill][Ill]
1 411 la[ 42 1 [ 2211 1
[ 2111 211.1 211[ 211
Table N-5 <6 13.3>
Table IV-5 <6 1 3 , s
38e.
Table IV-5 <6 1 3.3>
Table IV-5 <6 1 3 . 3
396. (2)
Table N-5 c6 13.5
Table IV-5 <6 1 3 3
40f.
Table N-5 <6 13.3.
Table IV-5 c6 13,s
Table IV-5 c6 13.5
46c.
Table IV-5 c6 I 3.3>
Table IV-5 <6 13.8
Table N-5 c6 1 3.3>
Table IV-5 <6 13.P
Sle.
Table N - 5 c6 I 3 . 5
53h
Table N-5e6 1 3,3>
[ 321 llx 6
[ 321 I
r 211[ 3 I,[3 I[ 3 1
I
Table IV-5 c6 13.P
Table N-5 c6 1 3.5
58b.
sea.
Table IV-5 e6 13,3>
Table IV-5 <6 13,3>
59b.
Table N-5 <61 3 3
Table IV-5 <6 1 3,3>
5%
Table N-5 c6 1 3 . 3
Table N-5 <6 1 3 3
61c.
Table N-5 c6 13.32
Table IV-5 c6 1 3.3>
62.
Table IV-5 c6 1 3 , s
[ui[ni [ luo(l1
1) I[ 211.[ 2111 211 13 11 211.1 21][111] 12111 211.1 3 11 211 [ 21111a].[ zi11211 12lII 211,[ 211[ 21) [ 211[ 211.[ Zllllll] 1211[ 211J 211[111] I 2111n1.11111[ 211 [ 211[1111,13 11211 [ 211[1111,[ 2111 211 [21111111.[ 2111 211 1211[1111J U][lll] [ 21111111,12111111] [211[1111.[1111[211 [llllI 211.1 2111 211 [11111211.[ 211[111]
I 1 42 1 0
1 42 I I 411 1 -3600
[ 411 I
I
33 1 1 321 1 1 321 I
2&?3
-225
o
-3600
-1003 -1003 1003
o
o
-UKX)
o
.m
o
o
o
-28110 WZ 45
0
0
0
-1003 24ll
100
0 -50 -WO
-3136 -1280
-2560
-225 1123
.IOU -7m 1936 -576 1280 720 1936
720
o
zm
XL?5 -XL?5
-pS
45
1125
1003 -1003 -24ll
0
0
-1800 4s
-7m 45
p5
-1600 UOo 1600
-1m
321 t~i z i i I [ 321 1 I 321 I
I 321 1
-MO o
-UOo 100
1800 450
.IW 450 -50 WO
-1280 320 -1280
-2304 64
320 1280
-PM
-1800 1125
0 0
MO UaO 0
ism
soo
uo
0 6UI -64
0
D
-1125
18M
0 MO
-1003 Z0
-64 64
ZmO 0
800 -8m 803
0 UU
-iw -800 0 UU 0
-1600 i w 1440 -1440
320
64
1280
64 .64 3136
320
0
-23M -1280 i m -64
-2304
1 3Zl 1 [ 3111 I
13111 1 [ Z?]
1440 1440 .IW
-576 -n6 576 IOU
-200 lam -200
o
0 .MO 0 0
0
0
uxa
o
640 0
1936
sm
200 -1003
-3m 576 1936 720
MO -100 1600
-1440 640
1280
0
0
-640
-1600
U60
0
0
ns
sm
sm
576 -576
1280 3Zal
8m 800
-720
-= 0 1280
-32.~1 1280
-lorn 18m -200 -1003 0 0
1003 1003 -1003
-MO
-
1003
rn
1003
[2211 1 [ W l l ] 720 45 7
0
1 1 ~ 0
2880
sm .ZW)
0 45
0
.llu
-50 lam
2 m
225
o
2Y)
qj
-3600 -1125 -1125
-yn
ZW)
45
-100 0
-50 0
.m
-225
1800
-1800
U O ~
-1600 .UOo
.awe 450
m
0
0
-1800 450
0
-3600
2025
Table IV-5 c6 1 3.3>
Table N-5 c6 13.3>
be.
Table IV-5 e6 I 3 . P
Table IV-5 <6 13,3>
Table N-5 <6 13.32
Table IV-5 <6 13.8
[Ill][Ill] [ 211[ 211.r 211[ 211 [ 211[ 211.[1111[ 3 I
[1111[3 I,[211[ 211
[3111 ]x[ 42 ]
Table N-5 c6 13,s
Table IV-5 <61 3 , s
Table N-5 <6 1 3 , s
Table IV-5 <613.5
=.(1)
Table IV-5 <6 1 3 . a
Table IV-5 c6 1 3 . 3
73d
Table N - 5 <6 1 3.3s
Table N-5 c6 1 3.3>
Table N-5 <6 I 3.3s
Table IV-5 <6 13,3>
Table N-5 c6 1 3 . b
Sle.
Slf.
Slh
Table TV-5 <6 1 3 . 3
Table N-5 c6 1 3 . 5
Table IV-5 <6 1 3.3>
Table IV-5 <6 1 3 . 5
Table IV-5 <6 1 3 3
Sf.
me.
Table IV-5 c6 I 3,3>
9od.
Table IV-5 <6 13,s
Table IV-5 c6 13.3>
Table N-5 <6 1 3 , s
Table N-5 <6 1 3,3>
Table N-5 c6 133.
94d
1211 1211 1 1-1
142
1 121113 1.111111 U l 12111211~12111 3 1 2111 a].[ 2ll[ 211 12111211.1 211[ 211 12111 211.1 21111111 12111211,111111 211 1211121].1111]1211 1 21111111.1 2111 211 121111111.[111][ 211 111111211.121113I [lllll 21l.l a ] [ 211 illllI 211.1 2111 a ]
1
0
l11111211.11111[ 211 111111211.1111lI 211
-
Wf.
I 2211 1x1 321 1 1 1 4 1 14111 I 4 1 1 1 1 3 3 1 13211 13211 13211 [ 3211 1 3211 13211 191111 131111 [ 2 2 1 [ P l l ] [ 2 U 1 ]
0 3XS
0 0 -2880 720 45 -720 45 3XS
-= 45 45 0 -1BM)
-3600 -225 -3600 0 0
o
.= 11s 0 -1121 -225 225 1125 1121 0 .I800
-low
-1600
-1000
25W
lorn
l a
0
0 0
u ~ )
-lo00 uX)
0 -low -50
-%
-wO
0
1800 1OW -1m
-25W
-2DI
-m
o lw
ZtO
18m 0 0
1800 4 0 -1m
%
100 XY) % lDOO
0
-1000 250 450 -50 22.50 250
-1280 320 -1280 0 -3136
a
-1280 -Pol
-a 64 320 1280
8W
am 8m 0 -2560 -1440 0 440 1440 -1440 -800 0
-2306
440
-64
1440 0 -1MO
zmo
0
0
6(0
320 1280 320 0 -1OU -576 -720 1936 576 -576 1280 72(3 1936 96 0
U60
-96 -576 96
-2m lam
.XY)
low w
.m
0
0 .lw
0 0 1600
lam
€dO
1024 0 .3u, 1936
m
XY)
.pWr
0
.no
-low
am sm
-320 1280 576 1936 720 1280 -2So 0
low
-100 -1600
.lea,
.yoo
4s
64
1440
a
leu,
-64
-1440
31% 0 i m
-P&$ .I280 i m 320
.a
-2560 0
sm
.leu,
.PO4
6(0
1280 320
0 800 -1606 0
uo 0
4wo
.low ism -200 -1WO 1MO 0 0
0
l w
720 4 -720
0 -11s 0
0 800
2880 0
0 0 3600 11s
-u,
-XY)
w
.XY)
PY)
-100 2MO 0 4 m
-y)
454
0 .lea)
o
a o ~ 0 45
.m 2025 lea, 0
-=
3606 -225 11s 11s zW .p5
0
Table N-5 c6 13,3>
Table N - 5 <61 3 . 3
95c.
Table N-5 <6 13.3>
Table N-5 <6 1 3,3>
97b.
Wc.
Wf.
Table N-5 c6 1 3.3>
Table N-5 c6 1 3 . 3
101b.
Table N-5 <6 13.5
103d
Table W-5 <6 1 3.3>
Table IV-5 <6 13,3>
lose
Table N-5 <6 1 3 . 3
107c.
1Mf. 1osa.I 2llll]x[ Z l l ]
Table N-5 c6 1 3 . 3
109e.
Table IV-5 <6 1 3.3>
116a.[llllll]x[ 321
I
Table N-5 <6 1 3.B
446
Tables of the SU(mn)>SU(m)xSU(n)
Coeficients of Fractional Parentage
References R.P. Bickerstaff, P.H. Butler, M.B. Butts, R.W. Haase and M.F. Reid, "3jm and 6 j tables for some bases of SUs and SU3", J. Phys. A15 (1982) 1087. J.C. Carter, J.J. Coyne and S. Meshkov, "SU(6) Clebsch-Gordan coefficients for the product 35 x 56" Phys. Rev. Lett. 14 (1965) 523. Jin-Quan Chen, "The SU(mn) > SU(m) x SU(n) isoscalar factor and S(f1 f2) > S(fi) x S(fi) isoscalar factor", J. Math. Phys. 22 (1981) 1. Jin-Quan Chen and Mei-Juan Gao, Reduction Coeficients of Permutation Group and Their Applications, (Science Press, Beijing, 1981). Jin-Quan Chen and Mei-Juan Gao, "A new approach to permutation group representation", J. Math. Phys. 23 (1982) 928. Jin-Quan Chen, Yi-Jin Shi, Da Hsuan Feng and M. Vallieres, "Symmetry and application of the SU(mn) > SU(m) x SU(n) coefficients of fractional parentage", Nucl. Phys. A23 (1983) 122. Jin-Quan Chen, D. Collinson and Mei-Juan Gao, "Transformation coefficients of permutation groups", J. Math. Phys. 24 (1983) 2695. Jin-Quan Chen, Mei-Juan Gao, Yi-Jin Shi, M. Vallieres and D.H. Feng, "Tables of onebody CFP for the group chains U(mn) > U(m) x U(n) and U(mp nqlmq np) > U(m/n) x U(p/q)", Nucl. Phys. A419 (1984) 77. Jin-Quan Chen, A New Approach to Group Representation Theory, (in Chinese) (Science and Technology, Shanghai, 1984). Jin-Quan Chen, Pei-Ning Wang, Zi-Min Lii and Xiong-Biaa Wu, Tables of the ClebschGordan, Racah and Subduction Coeficients of SU(n) Groups, (World Scientific, Singapore, 1987). Jin-Quan Chen, Group Representation Theory for Physicists, (World Scientific, Singapore 1989). Jin-Quan Chen, A. Novoselsky, M. Vallieres and R. Gilmore, "New approach to multishell calculations in multiple angular momentum coupling schemes," Phys. Rev. C39 (1989) 1088. C.L. Cook and G. Murtaza, "Clebsch-Gordan coefficients for the group SU(6)", Nuovo Cimento 39 (1965) 7095. R.H. Dalitz and A. Gal, "Strangeness analogue states in 9Be"' , Ann. Phys. 131 (1981) 314. J.P. Elliott, J . Hope and H.A. Jahn, "Theoretical studies in nuclear structure IV B", Phil. T h n s . Roy. Soc. (London) A246 (1953) 241. Mei-Juan Gao and Jin-Quan Chen, "The Clebsch-Gordan coefficients for permutation groups S(2)-S(6)", J. Phys. A18 (1985) 189. M. Hamermesh, Group Theory and its Application to Physical Problems (AddisonWesley, Reading, Mass, 1962). M. Harvey, "On the fractional parentage expansion of color-singlet six quark state in a cluster model", Nucl. Phys. A352 (1981) 301. K.T. Hecht and S.C. Pang, "On the Wigner supermultiplet scheme", J. Math. Phys. 10 (1969) 1571. K.T.Hecht, R. Le Blanc and D.J. Rowe, "New perspective on the U(n) Wigner-Racah calculus: I. Vector coherent state theory and construction of Gel'fand basesn, J. phys. A 20 (1987) 2241. C. Itzykson and M. Nauenberg, Rev. Mod. Phys. 38 (1966) 95.
+
+
+
The Algorithm
447
H.A. Jahn and V. van Wieringen, "Theoretical studies in nuclear structure IV. Wave functions for pshell, Part A. (pnIpn-',p) fractional parentage coefficients", Proc. Roy. Soc. A209 (1951) 502. H.A. Jahn, "Direct evaluation of fractional parentage coefficients using Young operators. General theory and (412,2) coefficients", Phys. Rev. 96,(1954) 989. Xiangzhu Li and Qianer Zhang, "Isoscalar factor of symmetric group chain Sn 3 S,,-l'l, J. Mol. Sci. (in Chinese) 5 (1987) 113. Xiangzhu Li, "On the symmetry properties of outer product reduction", J. Phys. A 22 (1989) 2299. Xiangzhu Li and J . Paldus, "Explicit representation of Gel'fand-Tsetlin states in Clifford algebra unitary group approach", Int. J. Quantum Chem. XXXVI (1989) 127. M. Machacek and T. Tomozawa, "SU(6) isoscalar factor for the product 405 x 56 + 56,70"', J. Math. Phys. 17 (1976) 458. V.A. Matveev and P. Sorba, "Quark analysis of multi-baryonic system", Nuovo Czmento 45 (1978) 257. M. Moshinsky, "Gel'fand states and the irreducible representations of the symmetric group", J. Math. Phys. 7 (1966) 691. I.T. Obukovsky, Yu. F. Srnirnov and Yu. M. Tchuvilisky, "On the construction of wave functions in the six-quark system", J. Phys. A 15 (1982) 7. G. Racah, "Theory of complex spectra I", Phys. Rev. 61 (1942) 186. G. Racah, "Theory of complex spectra 11", Phys. Rev. 62 (1942) 438. G. Racah, 'Theory of complex spectra 111", Phys. Rev. 63 (1943) 367. G. Racah, 'Theory of complex spectra IV", Phys. Rev. 76 (1949) 1352. G. Racah, Group T h e o y and Spectroscopy (Notes from Princeton, 1951). L. Schulke, "The SU6 Clebsch-Gordan coefficients for the reduction of 35 x 35 to 35 and 56 x 35 to 56", 2. Phys. 183 (1965) 424. S. Schindler and R. Mirman, "The decomposition of the tensor product of representations of the symmetric group", J. Math. Phys. 18 (1977) 1678. S. Schindler and R. Mirman, "Generation of the Clebsch-Gordan coefficients for Snn, Computer Phys. Commun. 15 (1978) 131. S.I. So and D. Strottman, "Wigner coefficients for SU(6) > SU(3) x SU(2)", J. Math. Phys. 20 (1979) 153. L.J.A.M. Somers, "Unitary symmetries, the symmetric group and multiquark systems", Thesis at the University of Nijmegen, 1984. D. Strottman, "Evaluation of SU(6) 3 SU(3) x SU(2) Wigner coefficients", J. Math. Phys. 20 (1979) 1643. D.Strottman, "Multiquark baryons and the MIT bag model", Phys. Rev. D20 (1979) 748. Hung-Zhou Sun, "On the irreducible representation of compact Lie group of rank two, I", Phys. Energ. Fortis Phys. Nucl. 4 (1980) 73. B.G.Wybourne, Classical Groups for Physicists (Wiley, New York, 1974). V. Vanagas, Algebraic Methods in Nuclear Theory (in Russian) (Mitinis, Vilnius, 1972). Zong-Ye Zhang and Guang-Lie Li, "Symmetry classification for excited states of hyper nuclei", Acta Phys. Sinica 26 (1977) 467. Qianer Zhang and Xiangzhu Li, "Coupling coefficients for the symmetric and unitary group", J. Phys. A20 (1987) 6185.
