Inner multiplicity of unitary groups — A modified version

Computer Physics Communications 44 (1987) 221—225 North-Holland, Amsterdam

221

INNER MULTIPLICITY OF UNITARY GROUPS

-

A MODIFIED VERSION

Samuel THOMAS Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario K7K 5L0, Canada Received 9 September 1986; in final form 30 October 1986

PROGRAM SUMMARY Title of program: IMUG1

counting all distinct Gelfand patterns which belong to the same weight [1].

Catalogue number: AATL Program obtainable from: CPC Program Library, Queen’s Urnversity of Belfast, N. Ireland (see application form in this issue) Computer: Honeywell DPS 8 or Any IBM PC Compatible Micro; Installation: Royal Military College, Kingston Operating system: Honeywell-CP6, Micro-MSDOS ver2.1 Programming language used: FORTRAN 77 No. of bits in a word: 36 for Honeywell, 16 for Micro

Restriction on the complexity of the problem The program as implemented here can handle SU(n) groups with a rank less than or equal to 29. Typical running time The multiplicity y(O, 0, 0, 0, 0) 5 in the representation D(1, 1, 0, —1, —1) is calculated in 0.31 s by the Honeywell computer and in 20 s by the microcomputer. The multiplicity y(O, 0, 0, 0, 0, 0, 0, 0, 0, 0) = 90 in the representation D(1, 1, 1,1,0,0, —1, —1, —1, —1) is calculated in 1.6sby the Honeywell computer and 295 s by the microcomputer. As expected if the multiplicity is a large number it takes more time to calculate it. The third example for n 30 given below took 0.5 s by the mainframe computer and 75 s by the micro.

Peripherals used: disk (hard and floppy) and terminal No. of lines in combined program and test deck: 470 Keywords: general purpose, Lie algebra, inner multiplicity, unitary groups, Gelfand patterns Nature of the problem To compute the inner multiplicity of a particular weight in a given representation for the special unitary groups using Gelfand patterns. Method of solution The inner multiplicity of a particular weight in a given representation, characterised by the highest weight, can be found by

Unusual features of the program There is no restriction on the dimension of the representation. Other programs [2] have restriction on the dimension of the representation. This program can be easily modified for higher rank algebras by adding and modifying certain source statements which are marked by comment lines. This program can also be used to obtain the various possible Gelfand patterns from a given partition. References [1] S. Thomas and M.T. Sunny, Comput. Phys. Commun. 14 (1978) 267. [2] B. Kolinan and R.E. Beck, Comput. Phys. Commun. 6 (1973) 24.

OO1O-4655/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

S. Thomas

222

/

Inner mu/tip/idly of unitary groups

LONG WRITE-UP 1. Introduction

multiplicity, denoted by YM(m). The multiplicity of the highest weight M is unity, i.e. YM(M) 1. For the SU(n) groups the (1 n 1) dimensional weight space is embedded in an n-dimensional Euclidean space R’1 such that the weights are n-dimensional vectors =

Lie algebras and their representation enter physics in a variety of ways. They have wide applicability such as in the various shell models that are used in physics, the atomic shell model, the nuclear shell model, molecular shell model, as well as in particle physics. The irreducible representation of Lie algebra plays a key role. These representations are classified by their characters and a knowledge of the inner multiplicity of weight is an important factor. There are many formulae to calculate the inner multiplicity of weights, but they are too complex for higher dimensional representations and for higher rank [1]. In a previous paper [2J we have published a program called IMUG which calculates the inner multiplicity of a particular weight in an irreducible representation of SU(n) using Gelfand patterns [3]. That program handles SU(n) groups with rank less than or equal to 9 and was written in BASIC-PLUS. There was no restriction on the dimension of the representation. The purpose of this program is to modify the previous program so that it can handle groups with a rank less than or equal to 30. Moreover, this program is written in FORTRAN77 and can even run using a PC compatible microcomputer. It is also possible to extend the program for higher rank groups by adding and changing a few source statements which are indicated on the source program by comment lines. •

•

.

.

.

.

2. Theory For purpose of completeness we describe here the theory behind the method. Let G denote a classical compact Lie group. An irreducible representation of G can be uniquely labeled by the highest weight of the representation, M (Mi, M2 M,), / being the rank of G. This representation is denotes by D(M). An arbitrary weight m (mi, m2,..., m1) may appear once or several times in the representation D(M). The number of times the arbitrary weight m appears in the representation D(M) is called the inner =

=

•

.

.

=

m

=

(m

m 2,...

)

m1, m

1

—

(1)

1+1

for which the additional condition holds m~+ m2 +

+

+

=

0.

(2)

(Here m k /(l+ 1), where k~ integer.) The character x ( M) of an irreducible representation D(M) of the group G is given as =

=

•

~(M)

~

=

•

e”~’

~M(~)

(3)

‘~,

m~D(M

where (m, (m,

q)

is the linear form

~~&) =

+ m2~11i2+

=

.

..

+m,,v,,. (4)

For any compact group it can be shown that V M(~ —

n~D(M)

/

L(rn),m’ —

~

=

y(Mf)yM(m~)

x ~

(5)

M’ m’GD(M’)

where 0 = =

—,

1, if m m and Th’ any weight in G’ 0, if m’ ~ =

and ~(M’) is the number of times the irreducible representation D(M’) of the subgroup G’ occurs in an irreducible representation D(M), if G is restricted to G’, and is known as the restriction multiplicity. Under the restriction a weight m of G goes over into a weight m’ L ( m) of G’. Eq. (5) can be simplified for SU(n) group by choosing GS G’ ® {U(1) ® ® U(1)}~_~ as a subgroup. This subgroup has the same inner multiplicity structure as G’. The weights of the subgroup G~are denoted by =

=

. . .

(m1,..., m2, Ar,..., X1 .) = (ms, A), (6) 5 = (mi,..., m~)is a weight of the subwhere m m

=

S. Thomas

group G’ and A,, I

=

/ Inner multiplicity of unitary groups

1,..., 1— s are the weights of

U(1) subgroups of Gs. Eq. (5) becomes yM(m)

=

(M1,..., M,,) is related to U(n) representation characterized by the m,,,’s through m1~=

(7)

~

223

=

—

M2

—

M~, (13)

W

where

m(~l)~ M~_1 M,,, =

m,,~=M~—M,,=0.

yM~(Ls(m))=yMS(Ls(m))

Further simplification is achieved by choosing SU(n 1) for G’. Then the subgroup GS of the group SU(n) is given by —

SU(/) ® U(1),

n

=

1+1, s

=

11.

=

1,

In the Gelfand pattern betweenness condition m,k÷1~ ~ ~

m.k

satisfies the usual (14)

(8)

Eq. (11) with the Gelfand pattern provides the

(9)

prescription to find the multiplicity y M(~ For a given weight m in an irreducible representation D(M) of SU(n), the subgroup content of D(M) with respect to SU(n 1) ® U(1) is considered. From the weight M (M1, M2,..., M~)the m,,,’s

Under this condition ~(Mt_i)

—

—

so eq. (7) becomes yM(m)

=

~

1

=

YMI_l(Lt_1(m))6L1_I(m)D(MI_1),

(10)

M’

with =

yM’1(L(m)).

=

1~f’1(~W) M’2(M’’)

?J’(t~f2)

6L1_1(m).D(M1_I)ôL~_2LF_~(m),D(M1_2) 000

eq. (11) it can be seen that only those irreducible representations D(M’1) of SU(n 1) ® U(1) containing the projected weight L’ ‘(m) contribute to the multiplicity M(~)~Continuing in this fashion until the subgroup SU(2) ® U(1) 0 U(1) is reached, the multiplicity yM(m) is equal to the number of all representations D( M’) reached in this manner which contained the projected weight L’... L’ ‘(m). —

If the last step is repeated for further subgroups we get

X6LIL2

can be calculated which will give the first line of the Gelfand pattern. The subgroup content D(M’’) of D(M) is given by all possible second lines of Gelfand pattern satisfying eq. (14). From

(11)

L°’(m),D(M’)’

.~..

It can be shown that along every line of the Gelfand pattern there is a unique set of sums,

where Mi=(Ml; A 1,...,A,~) where M’ is a SU(2) highest weight. The above equation could be related to the Gelfand patterns. A Gelfand pattern is given by I m1,,

m2,,

rn3, m2,,_1

[

•

•

•

rn,,,

1

n—i

S3

=

m, + i.~1

A,

=

m,,,,

where m

I

~-

~

k=1

j, =

I

=

(15)

1, 2,..., n,

(m1, m2,..., m,,) is the particular

weight of the representation D( M), which decides the projected weight.

mu

(12)

where the m,,, characterize an irreducible representation of U( n). The irreducible representation D( M~~) of SU( n) with highest weight M” —1 =

Implementation 3.1. Input ~•

This program can be run interactively. Since the program is written userfriendly, the user is

224

S. Thomas

/

Inner multiplicity of unitary groups

prompted with questions. In answer to the question “TYPE N = L +1 OF THE ALGEBRA”, the user inputs an integer value less than or equal to 30. Here L is the rank of the algebra. The second input to the program is the highest weight of the representation. Each component of the highest weight can be expressed in the form k,/n where k1 is an integer, only the k, values are given. So in answering the question “ENTER THE HIGHEST WEIGHT OF REPRESENTATION” the user inputs the k, values separated by commas. The third input to the program is the particular weight whose multiplicity is to be determined. In answer to the question ‘ENTER THE PARTICULAR WEIGHT’ the k. values for the particular weights are given separated by commas. 3.2. Output There are two options available for output. The choice is indicated by selecting ‘Y’ (Yes) or ‘N’ (No) to the question ‘DO YOU WANT ALL POSSIBLE GELFAND PATT’ERNS TO BE PRINTED?’. If the choice is Yes (type in the letter ‘Y’) all the possible gelfand patterns will be printed out as well as the multiplicity. If the answer is No (type in the letter ‘N’) then the output contains the multiplicity. 3.3. Size of the problem The program handles SU( n) group with n ~ 30. The text of the program contains comments that will show the statements that will be changed to handle higher rank algebra than 30. Since the multiplicity of each weight is calculated independently of other weights there is no restriction on the dimension of the irreducible representation.

3.4. Algorithm for inner multiplicity Step 1. The highest weight vector (input) M = (M3, M2, M~) is converted to m3~, m2~ ~ ~ = 0, using eq. (13). Step 2. Form A = 1m,,, Step 3. From the particular weight vector (input) m = (m1, m2,..., m~) form the set of numbers (~, S~)using the formula .

• .

,

~.

~

J

S~.=~ m,+ LA,

wherej= 1,2, n—i.

Step 4. Set Flag = 1; COUNT = 0. Step 5. If Flag = 0 then STOP. Step 6. Determine a set of numbers satisfying the conditions m1~~ m1~1~ m~,,~ m2~~ ~ ms,, ~ ~ ~ ~ 0, and m1 ~

+ m 2n -1 +

COUNT

=

+m

-

=

S~_1,

COUNT + 1

Step 7. Store the elements in an array satisfying the conditions in step 6, and set Flag = Flag + 1 if tracing forward (going from root to branches of the tree) or Flag = Flag 1 if tracing backward (going from branches to root). Step 8. GO TO STEP 5. —

References [11B.

Kolman and R.E. Beck, Comput. Phys. Commun. 6 (1973) 24. [2] S. Thomas and MT. Sunny, Comput. Phys. Commun. 14 (1978) 267. [3] R.M. Delaney and B. Gruber, J. Math. Phys. iO (1969) 252.

S. Thomas

/

Inner multiplicity of unitary groups

225

TEST RUN OUTPUT FORTRAN IMUGi OVER cc

RUN

FORTRAN 77 VERSION COl 43.100> 44: 431.000> 430: ERRORS FOUND 436.000> 451.000> ERRORS FOUND

5: 20:

455.000> 470.000> ERRORS FOUND

4: 19:

JUL 10 ‘86 INTEGER IDIM,MDIM END

LINK COO here * SHARED_COMMON. :SYS (Shared Library) associated. *

: 0

TOTAL ERRORS FOUND: 0

: 0

SUBROUTINE 5ORT(D.N1.IDIM) END TOTAL ERRORS FOUND: 0

: 0

SUBROUTINE PR(N1,A1TEMP.S9.A1,IDIM) END TOTAL ERRORS FOUND: 0

*

No linking errors. Total program size

9K.

INPUT DATA r TYPE N=L+1 OF THE ALGEBRA

RUN LINK COO here * SHARED_COMMON. : SYS (Shared Library) associated. * *

No linking errors. Total program size

?1O ENTER THE HIGHEST WEIGHT OF REPRESENTATION ?10,1O,10.10.0,0.-10,-1O,-10.-1O ENTER THE PARTICULAR WEIGHT ?0.O.O,O,O,O,0.0,0.0 DO YOU WANT ALL POSSIBLE GELFAND PATTERNS TO BE PRINTED? ? ‘N’

9K.

INPUT DATA OUTPUT TYPE N=L+1 OF THE ALGEBRA

ENTER THE HIGHEST WEIGHT OF REPRESENTATION ?5.5,0,-5.-5 ENTER THE PARTICULAR WEIGHT 70,0,0,0,0 DO YOU WANT ALL POSSIBLE GELFAND PATTERNS TO BE PRINTED?

HIGHEST WEIGHT ( 10 10 10 PARTICULAR WEIGHT ( 0 0 0 MULTIPLICITY IS: 90 CSTOPa

10 0 0 -10 0 0 0 0 0

-10 -10 -10 0 0 )/ 10

)/

10

RUN LINK COO here * SHARED_COMMON. :SYS (Shared Library) associated. No linking errors. * Total program size = 9K.

OUTPUT

INPUT DATA

GELFAND PATTERNS TYPE N=L+1 OF THE ALGEBRA SUM 4 ELEMENTS

(

0

1

1

2

SUM3ELEMENTS(1 SUM 2 ELEMENTS ( 1 SUM 1 ELEMENTS 1

1 1

1)

SUM 3 ELEMENTS ( 0 SUM 2 ELEMENTS ( 1 SUM 1 ELEMENTS 1

1 1

2

SUM 2 ELEMENTS C 0 SUM 1 ELEMENTS 1

2

ELEMENTS C 0 ELEMENTS ( 0 ELEMENTS C 1 ELEMENTS I

0 1 1

)

SUM 2 ELEMENTS C 0 SUM 1 ELEMENTS 1

2

)

)

730 ENTER THE HIGHEST WEIGHT OF REPRESENTATION ?5O,50.50,5O,50,-10,-10,-10,-10,-1O,-1O,-1O,-1O,_1O,-IO,_1O,_1O,_ 10.-10.-10,-10,-1O,-10.-1O.-1O,-1O.-1O,-1O,-1O,-1O

SUM SUM SUM SUM

4 3 2 1

)

ENTER THE PARTICULAR WEIGHT ?20,20.20,20.2O.20.20,20.20,2O.-1O-1O-1O-1O-iO-1O-1O-1O. 10.-10.-1O.-1O,-10.-10.-1O.-10.-1O.-1O.-iO.-1O DO YOU WANT ALL POSSIBLE GELFAND PATTERNS TO BE PRINTED? ? ‘N’ 2 2

2

)

OUTPUT

HIGHEST WEIGHT -10 -10 -10 -10 -10 -10

C

50 -10 -10

PARTICULAR WEIGHT HIGHEST WEIGHT C ~ 5 PARTICULAR WEIGHT C 0 MULTIPLICITY IS: 5 *510P*

0 0

-5 ~ ~.‘ 0 0 0 )~‘ ~

(

-10 -10 -10 -10 -10 -10 -10 -10 MULTIPLICITY IS: 42 *STOP*

50 50 50 50 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 -10 )/ 30 20 -10 -10

20

20 -10 -10

20

20

-10 -10 )/ 30

20

20

-10

20 20 -10

-10

20 -10

-10 -10