Weight multiplicity for cartan classes

Computer Physics Communications 29 (1983) 201—209 North-Holland Publishing Company

201

WEIGHT MULTIPLICITY FOR CARTAN CLASSES V. AMAR, U. DOZZIO and C. OLEARI 1stituto di Fisica dell’Università, 43100 Parma, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy Received 26 July 1982

PROGRAM SUMMARY Title of programs: BCD DAM and MAIN

Card punching code: BCD

Catalogue number: ACED

Keywords: general purpose, representation, degeneracy, Cartan classes, Lie algebra, weight

Program obtainable from: CPC Program Library, Queen’s Urnversity of Belfast, N. Ireland (see application form in this issue) Computer: CDC Cyber 730; Installation: Centro di Calcolo Interuniversitaro deil’Italia Nordorientale, Casalecchio sul Reno, Bologna, Italy Operating system: NOS/BE Programming language used: FORTRAN IV High speed storage required: 79000 words for test run

Nature of the physicalproblem The importance of simple Lie groups in physics is well known. This program computes the multiplicity of the dominant weights in a given irreducible representation of the Cartan classes B(N),C(N) and D(N). Method of solution Given the rank N the program BCD DAM generates the program MAIN. This last program computes the multiplicity of the dominant weights in a given irreducible representation of a group G(N) for the classes of Cai-tan B(N), C(N) or D(N) by the reduction G(N)—~G(N—l)®U(l)--... —~G(2)®

No. of bits in a word: 60 Peripherals used: card reader, line printer No. of cards in combined program and test deck: 809

Typical running time Compile time was 12.4 s, total execution time for the test run was 6.254 s.

OO1O-4655/83/0000—0000/$03.OO © 1983 North-Holland

202

V. Amar et a!.

/

Weights multiplicity for Cartan classes

LONG WRITE-UP

the JR of G(N 1) ® U(l) defined by M~’ (M[~’,..., M~jl’,mN)in theIR of G(N) defined —

by 1. Introduction

=

This quantity

MN.

can be G(N-I)®U(I)

evaluated by the algorithm given by Mickeisson [4]

Finite-dimensional irreducible representations (JR’s) of simple Lie algebras play an important role in physics. The computation of the inner multiplicity of these representations is a classical problem, that has general methods of solution [1,2], which are too complicated and consequently not very effective. This is confirmed by the fact that the existing computing programs [3] using these general methods cannot deal with JR’s with more than 1000 weights and rank greater than 9.

for the simplectic group and by our algorithms [5,6] for orthogonal groups. By applying iteratively (N 2) times relation (I) we obtain —

M~N,

Y(m)

=

...

...

1cM’~ M~2cM~

M”~

M’~’~Dm

M~tCMA

M2c~

M~1Drn

M2~ir

M~2Dm

M~N~

M~~iLI)®~I)

~ K

Using the results of recent work [4—6],whose aim was the evaluation of the branching multiplicity in the reduction G(N)~ G(N— 1)® U(l), where G(N) is a simple Lie group of rank N, we give a very powerful computing program for the

~MG

M3

N—K ~U(I))

-

N-K+I G(K-I)(U(l))

M2

N—3

G(3)a~U(I))

YM2

N-2

N-2 I))

)‘(m)

(2) M~( 2)~U(I))N2

Cartan classes B(N), C(N) and D(N). The class A(N) is not included since a very effective program is already available for this case [7].

where the quantity Y(m) is 1 if G(2) SO(4), while if G(2)= SO(S) or Sp(4) it can be evaluated by formula (2) of ref. [10] (one must recall the isomorphism Sp(4) SO(5)). To any element of the sum (2) we associate the following

2. Algorithm description

patt~

Let us consider an IR of the group G( N) belongingtoaCartanclassB(N), C(N)orD(N), defined by the highest weight MN

=

(Mr, Mr,..., Mfl

(the upper index is not a power). Consider a dominant weight m

=

(m1, m2

=

M[~

M’

M1~’

M~’

M~

M~

M~

MI

M~1 M~’~ N— n~v

...

M~

m2

mK+

m~

‘liv

I (3)

mN).

The components of these weights are given by the standard labels of Racah [8]. The inner multiplic-

whose rows must satisfy the constraints of the sum (2) and be such that any ~ * 0. In the next section

ity of the weight m in the JR of G(N) defined by M~’can be given by the following formula [9]

we give explicit evaluate the ~ ‘s. rules to generate patterns and to

(1)

3. Computer program description The computer program is orgathzed as follows:

is the branching multiplicity of

Corresponding to any givenare Cartan class, rank JR the dominant weights generated. For and any

=

~

ML,>

~.MGJJN~)

MN - CM N YMGJN -I) M~’~Dm MN where ~M~i_’~? G(N—l)~U(l)

U(

®U(I)

~( m)

/

V. Amar et aL Tape 2

Work By Program BCD DAM

Tape 3

Input

4-....

~3O2co,,, nUn,,

hi ~hcst ranI~

I ~.

—+4Generate subroutine URN 1

—

[REWIND Tape 3 Transfer from Tape 2 to Tape 49 instructions cards

—

4— —

—

a ~ REWIND Tape 3 ~ Generate subr.

—.4 Generate

—

4— —

.-,

CEN 3 ,3=2

N

DINB N

4—

a ~ REWIND Tape 3 z Generate subr. GF K K ,K3 REWIND Tape 3 Transfer from Tape 2 to Tape 21 instructions cards

—

4—

~IREWIND Tape 3 ~,

—

flGenerate subr.

4—

9’ —

REWIND Tape 3 a Generate subr. CF 3 K 3=1,2 K— K—3

.Subr.

URN 3

F

~Subr.

DINB N

F

‘Subr.

DINB 3

~-

~Subr.

CF 2 2

,Subr.

CF K K

J.

3

I

N—*

Subr.

CF K—i K

3

.Subr.

,

CF 3 K

N—i

F

....4_{~nerate subr. DEC

—

I—

I N—i

CF K—i K, K=3

REWIND Tape 3 Transfer from Tape 2 to Tape 56 instructions cards

—

GEN 1

3

‘I’

—

Subr.

1 N—i

subr. CF 2 2

REWIND Tape 3 Transfer from Tape 2 to Tape 15 instructions cards

—

F

3

5 ~REWINDTape3 Generate subr. DINB J ,3=i

—~~4 Generate

—

‘Program MAIN

3

‘I,

REWIND Tape 3 Transfer from Tape 2 to Tape 29 instructions cards

—

—

4

instructions

[REWIND TAPE 2 ___*{ Generate Program MAIN

—

—

Tape

WRIfl~on Tape 2 392 Computing instructions of input

—

Input I N

—

Weights multiplicity for Cartan classes

Subr.

DEC

‘4, —

REWIND Tape 3 Transfer from Tape 2 to Tape 10 instructions cards

—

4—

—

—

~ REWIND Tape 3 ~ Generate subr. INT 3 ,3=1,2

—

—+-4

—

—4-I_Generate sub,,. TRANS

Generate subr.

VF

Fig. I. Structure of program BCD DAM.

3

N—iJ- ~Subr.

INT 3

}-

,Subr.

VF

~-

~subr.

TRANS

203

204

V. Amar et aL

/

Weights multiplicity for Cartan classes

dominant weight the related patterns are generated, then the corresponding contribution to degeneracy is evaluated. Since any JR of SO(2 N + 1) can be reduced to a direct sum of JR’s of SO(2N) with branching multiplicity equal to 1, for the class B( N) of Cartan we choose the following reduction chain so(2N + 1) — SO(2N) so(2N — 2) ® u( 1) —*

-~ so(4)® (U(l))”~2, so we use the patterns related to the class D(N) instead of that of the class B( N). ...

—‘

Input

i

MAIN

I I I

TRANS

1

~

I

+4 4

~,

=

I

GEN1

I

GEM 2

‘N VA

GEN N~F-±

The numbers of sums in relation (2) (hence the number of the pattern elements and of the weight components) depends on the rank N; therefore the program also depends on the rank. The proposed program scheme consists of a first program, named BCD DAM, which, for fixed highest rank N, generates by transfer from tape to tape a second program for the computation of the inner degeneracy in any JR of the classes B(K), C(K) and D(K) with 2 ~ K~N. The structure of program BCD DAM is shown in fig. 1, and at the end of its run tape 4 contains the program for computing the inner multiplicity. The structure of this second program is given in fig. 2: a) Program MAJN reads the input. b) The subroutines GEN I, with J 1,2,..., N, generate the components of the dominant weights by requiring the following constraints: i) dominance condition: for the classes B(N) and C(N)

Output

~ ~1

—

mI>~m 2>~...>_mNI>_mN>O;

2

1

!~i DINBI.

I

—

for the class D(N)

1’4 2!~+4

I I

‘t

2

~

DINB N

CF

N—i

N—i

çF

N—2

N—i

y,4

+4

3 VA IGF 3

3

LGF

3

i

IGF

2

2

[GF

INT

I I

J

MN=(Mj~~~,M~M~) -

for the classes B(N) and C(N) K ~(M,N_me)~0VK=1,2

N;

1=1

— for the class D(N) i

i

K

~(M7”_m.)~0VK=l,2

N,

i=I

__________

lINT

~ ~ImN_1~ImNI. ii) condition for a dominant weight of belonging to the weight diagram of the IR defined by the highest weight

2

*t IDES +4 I

I

2 N-I

I

+4 INTN—I

+4

I

IVF

Fig. 2. Structure of the program on tape 4 at the end of the run of program BCD DAM. Path I for class B(N), path 2 for classes C(N) and D(N).

i—I

Note that: — for the class B(N) all the numbers rn, have to be integer or half-odd like the ~jrN ‘s. — for the class C(N) the numbers rn are integer, like the

V. Amar et aL / Weights multiplicity for Cartan classes MN

‘s, and must satisfy the additional condi-

tion

e) Finally, subroutine DEG, using subroutines INT J, with J am 1,2,. (N — 1), and subroutine VF, evaluates the quantities . . ,

N

(M,” — m,) = even integer.

~ ~

205

related to any pattern. These branching multiplicities are equal to the num-

i

y~QK ‘~ (K) e U( I)

— for the class D(N) all the numbers rn, are

integer or half-odd like the M1” ‘s, and must satisfy the additional condition

ber of possible sets {q1) which satisfy: for the class C [4]

—

N

~ (M1”

—

m,)

=

even integer.

K+l

2

~

K+I am

~

K MK+t + ~ ~K

+ mK+I,

(4)

i= I i—I

The subroutine GEN N also prints the output. In the output the weights are also written in the basis of the fundamental weights and the translation between the two bases is made by subroutine TRANS (see appendix). c) The subroutines DINB K, with K = 1,2... N, work only for IR’s of the class B(N) and

i—I

J(

Mi~ ~‘ ~ max(M~ 1, M[’), mm( M’~ M~ ~ ~ max( M,~’,M,”)

L

min(M~~,Mfl

~)

~,

~ 0;

~ qK+I

(4a)

for the class D [5,6,12]

—

generate the whole set of JR’s of class D(N), which are contained in the given IR of B(N) and contain the weight m. The JR’s of class D(N) are obtained by imposing the conditions required by the Gel’fand triangle [11] (i.e. the JR of D(N) must belong to theIR ofB(N)) and the conditions like ii) (i.e. JR’s of D(N) must contain the weight rn). d) The subroutines GF J K, with Jam l,2,...,K and K = 2,. (N — 1), excluding J = 1 with K 2, generate the pattern elements in such a way that the weight (M!’, ~ mK+I,...,mN)

i= I

K+l

2 ~ q ~

JI

K+1 =

~

M~

K

+ ~

M,”

+ mK÷ 1~

(5)

M(~’~ q~~ max(M~’, Mfl, min(Mf±1, M~)~e q1 ~ max(M~’, Me”)

I

I q~÷~ =mK+l,

(5a)

. . ,

where I MxK+~111 ~ Me’. member that [5]

For M~< 0 we re-

.M~ +M~,+mK+I~YM[’

1) is a weight belonging to the JR of the group G(K + 1) ® (U(l))N~N~~, whose highest weight is (M~~’,Mr~’,...,

We mentioned that in ref. [5,6] the proposed algorithm for the evaluation of j~ ‘s in the class D stood up to a very large numerical test, while recently [12] we gave the proof of its

MKK+l ‘ MK~ K+ I’

mK÷2,...,mN),

2) is the highest weight of an IR of the group G( K) ® (U( I ))N - K which contains the weight m. In order to satisfy the above conditions I) and 2) we must use rule ii) and the constraints 4a) and 5a) given below, which assure that

validity. Subroutine DEG also 2

Y( m)

4. List of common variables N

YM~K)~U(I)*0.

evaluates the factor

N—2

am

highest rank,

NN = rank of the considered IR ~ N,

206

V. Amar et a!. / Weights multiplicity for Cartan classes

KLASS = class of Cartan (KLASS = 2,3,4 for the classes B,C,D), IDEG am y~, CQ contribution to the degeneracy due to a pattern row, FM(NN) highest weight, F(I,J) for J ~ I = pattern element, F(I,J) for J < I = F(K,I — 1), W(NN) dominant weight in the Racah nomenclature, JW(NN)= dominant weight in the basis of the fundamental weights, SW(K) am W(I), SF(K) I F(J), Q(I) variable q. in the relations (4) and (4a) or (5) and (5a), V(I) lower bound of the variable Q(I), R(I) upper bound of the variable Q(I). am

am

am

am

am

am

At the end of a run of program BCD DAM, program MAIN is available with all its subroutines on tape 4. Jnput to program MAIN consists of the following cards: 1 card, with the rank NN of the considered IR and the Cartan class B, C or D written in FORMAT(13,Al). NN cards, any one of which gives a component of the highest weight defining the IR in FORMAT (2X,F5.l). The highest weight is given by the Racah labels and the components are ordered in decreasing sequence. For the next computation other cards are needed, which are ordered as in the previous case. Blank card stops the run. —

—

—

—

am

5.2. Output 5. Input—output description

The output consists of a row, in which is printed the class and the rank, and a table, in which the numbers of the first column are the degeneracy of

5.1. Input Jnput to program BCD DAM consists of 392 instruction cards in the following order: — 43 cards, whose instructions are needed for making program MAJN, — 7 cards for subroutine GEN 1, — 49 cards for subroutines GEN J, with J = 2 N, 10 cards for subroutine DINB N, 29 cards for subroutine DINB I, with ~ 1,2,. (N — 1), — 65 cards for subroutine GF 2 2, — 15 cards for subroutines GF K K, with K = 3,... ,(N — 1), 21 cards for subroutines GF K — 1 K, with K ~ ,(N — 1), 56 cards for subroutines GF J K, with J = 1,2,...,(K —2) and K 3,...,(N — 1), — 56 cards for subroutine DEG, 10 cards for subroutines INT J, with J 1,2,. (N — 1), 20 cards for subroutine VF, 11 cards for subroutine TRANS. — 1 card, where the highest rank N is written in -

—

am

—

. .

—

am

—

the dominant weights, whose components are written in the other columns. The weights are shown both by Racah labels and in the basis of the fundamental weights. In tables are given the outputs for same small representations and low ranks, as the output of test runs. In order to give an idea of the power of the algorithm and of the program we report here a meaningful example of a computation: with a run of 200 s we obtained the multiplicity of 130 dominant weights of the JR (9,9,9,9,4,4,0,0,0,0) of SO(22), and, for instance, the dominant weight (9,9,9,8,1,0,0,0,0,0,0) has multiplicity 5869. Moreover the program BCD DAM was able to generate in 136 s program MAIN and its subroutines for rank 21.

am

am

—

. . ,

Acknowledgements

— —

FORMAT(J3).

The authors are grateful to Prof. B. Kolman and to Prof. R.E. Beck for a private communication.

V. Amar et aL / Weights multiplicity for Cartan classes

G.

Appendix

Racah, in: Group Theoretical Concepts and Methods in Elementary Particle Physics, ed. F. Gursey (New York,

For completeness we give the relation between the basis of rn’s and ~.L1’s (the ~~1’sare the components of the weights in the basis of the fundamental weights) am

rn1..

—

rn for i am 1,2,..

I mN — 1LN_Iam~

.,

N

—

2,

rnN for the classes B (N) andC(N), m~,,11+ mN for the class D(N), —

I2rn,.,~for the class B (N), I-EN

207

rnN mN

for the class C(N), I mN for the class D( N). —

1962). [2] N. Jacobson, Lie Algebras (Wiley, New York, 1966) p. 261. [3] B. Kolman and R.E. Beck, Comput. Phys. Commun. 6 (1973) 24. [4] J. Mickelsson, Rep. Math. Phys. 3 (1972) 193. [5] V. Amar, U. Dozzio and C. Oleari, Physica 114 (1982) 341. [6] V. Amar, U. Dozzio and C. Oleari, Nuovo Cimento 70 A (1982)460. [7] V. Amar, U. Dozzio and C. Oleari, Comput. Phys. Cornmun. 14(1978)413. [8] 0. Racah, Group Theory and Spectroscopy, CERN 61—68 (March 1961). [9] R.M. Delaney and B. Gruber, J. Math. Phys. 10 (1969) 252. [10] V. Arnar, U. Dozzlo and C. OleaH, Lett. Nuovo Cirnento 18 (1977) 13. [111 I.M. Gel’Fand, R.A. Minlos and Z.Ya. Shapiro, Represen-

References 11] N. Jacobson, Lie Algebras (Wiley, New York, 1966) p. 247.

tations of the Rotation and Lorentz Group and their Applications (Perganion Press, London, 1963) p. 353. [12] V. Arnar, U. Dozzio and C. Oleari, submitted for publication.

208

V. Amar et aL / Weights multiplicity for Carran classes

TEST RUN OUTPUT hI 5) MULTIPLICIT1 1 1 ~ ~ I ~ 1 ‘~

4

1’~ ‘.~

~ cU

ci b.i b L>

11 11 U0 .)7 ‘.u tI 0.)

1+U 147 1UL’ 103 ?1’~ 2c~ 4ci 44~

WEIGHT 4.p 4.0

(QACAH NOMENCL4TIJPE) 2.0 1.0 0.0 0.0 2,0 0.0 0.0 0.0

w~I~.1T(FU~JDA-•ItNT~LWtIGHTS ~ 1 1 I) (I 2

2

U

()

U

4.0 4.0 4.fl 4.0 3.0 3.0

1.0 1.0 1.0 0.0 3.0 3.0

1.0 1.0 0.0 0.0 1.0 0.0

1.0 0.0 0.0 0.0 0.0 0.0

U.0 0.0 0.0 0.0 (1.0 0.0

~

0

0

1

~

‘j

1

0 U

1

U

(I

I)

0 U C)

2

1

1,

j

u

~

:3.0 3.0 3.n

2.0

0.0 1.0 0.0

0.0 0.0 0.0

1 1 1

i’

c~ Ii

(C

1

C J

2.0

2.0 1.0 1.0

1

(C

U

3.o 3.0 3.fl 3.0 3.n 3.0 2.0 2.0 2.0 ~.o 2.0 2.0 2.0 2.0

2.0 1,0 1.0 1.0 1.0 0.0 2.0 2.0 2.0 2.0 2.0 2.0 1.0 1.0

0.0 1.0 1.0 1.0 0.0 0.0 2.0 2.0 1.0 1.0 1.0 0.0 1.0 1.0

0.0 1.0 1.0 0.0 0.0 0.0 1.0 0.0 1.0 1.0 0.0 0.0 1.0 1.0

0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 0.0

1 7

2

U

Ci

C)

U U 3

ii

1

2

u

1

U

1

C)

1 U 3

U

2.0 7.0 2.0 1.0 1.0 1.0 1.0 1.0 0.0

1.0 1.0 0.0 1.0 1.0 1,,0 1.0 0~0 0.0

1.0 0.0 0.0 1.0 1.0 1.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0

1 1

U

~

Ci Ci C)

2,,o

~ 4 (1 Cl

2 2

2 ~ ) Cl

n Cl

() 0 1 1

1)

0 0 0 C)

1 1

U IC U Ci

U

(C

U

1

1

CC

U

2

))

U

1 1 1 2 0

1) 0

C)

1

1 0

CC

U

U U

U 2

C)

1

U 1 0 0

I Ci U Ci 0

1 (3 U 3

U

1

CC

I)

Ci

(C U C)

ii U 2

1 1

(C

C) IC CC

0 U ii

U

Ct b) MULTIPLICITY 1 ‘~

b

~ 4 12 b 2~ b

~ 31 11 ‘~

1~Cb bb lSb 332

WEIGI-T

(PACAR NOMENCLMTUPE) 2.0 1.0 0.0 0.0 4.() 1.0 1.0 1.0 0.0 4.(~ 1.0 ~ 0,0 0.0 3.0 3.0 1.0 0.0 0.0 3.)) .0 2.0 0.0 0.0 3.() 2.0 1.0 1.0 0.0 3.o 2.0 0.0 0.0 0.0 3.0 1.0 1.0 1.0 1.0 3.0 1.0 1.0 0.0 0.0 3.0 0.0 0.0 0.0 0.0 2.0 2.0 2.0 1.0 0.0 2.0 2.0 2.0 1.0 0.0 1.0 1.0 0.0 2.0 1.0 1.0 1.0 0.0 ‘+.~

2.0 1.0 1.0 1.0

1.0 1.0 1.0 0.0

0.0 1.0 1.0 0.0

0.0 1.0 0.0 0.0

0.0 1.0 0.0 0.0

C~tIGHT(F UNC)A-CrNT.~L ~ 7 1 C) Ci 1 3 U 1 Ci 1 1 U U U 0 7 1 0 U 0 2 0 C) 1 1 IC 1 1 1 2 Ci 0 U 7 0 Ci 0 I 7 0 1 ~) U 1 u 0 0 fl I) 1 1 0 0Cl 1 01 0 01 1 0 0 1 0 1

1

U

fl

u

Cl 1

0 0

0 1 0

0 0 ()

0

))

1 U 0

‘‘ItNLL,~T’U-y.)

V. Amar et aL / Weights multiplicity for Cartan classes

209

DI 5) MULTIPLICITY 1

WEIiHT 3.5

(RACAH NOMENCLMTURE) 2.5 1.5 .5 .5

WEIc,.iT(F UNDA-~ENTAL WEIGHTS 1 1 1 1 0

3,5

2,~

.5

.5

—.5

1

1.5 1.5 1,5 2.5

1.5 1.5 .5 .5 2.5

1.5 .5 .5 .5 .5

4

2.5

2.5

1.5

1.5

b In

2.5 2.5

2,5 2,5

1.5 .5

.5 .5

.5 —.5 .5 —.5 .5 .5 —.5 .5

2 2 p ~

2

3.5 3.5 3.5 3.~ 2.S

b

2.5

1.5

1.5

1.5

14

2.5 2.5 2.5 1.5 1.5

1.5 1.5 1.5 •5 1.5 1.5

1.5 1.5 .5 •5 1.5 1.5

1.~

1.5

1.5

1.5 1.5 .5

1.5 .5 •S

.5 .5 .5

‘~

~ Y lb

.02 Sb

lob 56

24 Yb

i7~ 293 4b~

2.5

.5

2

0

0

1

0 0 1 0 o 0 ~) 1 0 1 (1 2

0

0 2 0 1 0

2 0 1 0 1 2 1) 1

1 1 ~ 1

1.5

1

0

0

3

0

1.5 —.5 .5 .5 .5 —.5 .5 .5 1.5 .5 1.5 —1.5 .5 —.5 .5 .5 .5 —.5 .5 .5

1 l 1

0

0

1

~

0

1

1

0

1 (1 0 0 0 1 0 0

0 0 0

0 1 2

1

0

o 0 0

1 Cl

1

0

NOMEHCLAT)H~t)

0

1 1 0

U

1

0

o

i

1 0 0 0

0 1 0

1 0 1

1

0

b( 3)

MULTIPLICITY 1 1 ~ 1

WEIGhT

(RACAH NOMENCLATURE)

2 2

a 1

2

2.0

0.0 1.0

3 0

0 1

0 2

2.0 1.0 1.0 0.0

0.0 1.0 0.0 0.0

2 0 1 0 u

0 2 0 (1 2

1 Ii 0

0 (C 0

1.0 1.0 0.0

3 5 6

~

1

wEIr,HT(F (JODAMENTAL WEIGHTS

1.0 0.0

3.0 3.0 3.0 2.0 2.0 2.0 2.0 2.0

1.0

1.0

1.0

IU

1.~

1.0

0.0

(1 1 1 2 ~ C)

14

1.0 0.0

0.0 0.0

0.0 0.0

1 0

lb D( 3) MULTIPLICITY 1 1 1 2 1 -

3 3 4 ~ 6

WEIGhT

4.0 4.0 3.0 3.0 2.0 2.0 2.0 2.0 2.0 1.0 0.0

(RACAH NOMENCLATURE)

1.0 1.0 0.0 0.0 2.0 1.0 1.0 0.0 2.0 2.0 2.0 0.0 1.0 1.0 1.0 —1.0 0.0 0.0 1.0 0.0 0.0 0.0

0

WEIGHT (F ))N1)A)’t.NTAL WEIGHTS

3 4 1 2 0 0 1 1 ~ 0 n

2 1) 3 1 4 2 2 0 o 1 0

NOMENCLATURt.)

0 () 1 1 0 2 0 2 0 1 0

NOMENCLATLJRt)