Reduction of the Kronecker products for rotation groups

HAL Id: jpa-00206829

https://hal.archives-ouvertes.fr/jpa-00206829

Submitted on 1 Jan 1969

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Reduction of the Kronecker products for rotation groups

P.H. Butler, B.G. Wybourne

To cite this version:

P.H. Butler, B.G. Wybourne. Reduction of the Kronecker products for rotation groups. Journal de

Physique, 1969, 30 (8-9), pp.655-664. �10.1051/jphys:01969003008-9065500�. �jpa-00206829�

655

REDUCTION OF THE KRONECKER PRODUCTS FOR

ROTATION

GROUPS

(1) By

P. H. BUTLER and B. G.

WYBOURNE,

Physics Department, University ^of Canterbury, Christchurch, ^{New Zealand.}

(Reçu

^{le 14 mars}

1969.)

Résumé. 2014 Des méthodes pour

analyser

^la réduction des

produits

Kronecker pour ^les groupes de rotation

orthogonale

^{en utilisant} des méthodes fonction S de D. E. Littlewood sont

développées.

^Les

propriétés

^des

représentations

^{de la rotation sont discutées sur certains}

points

^ainsi que la méthode de différence des caractères utilisés dans le cas des groupes ^de rotation unidimensionnelle. Ces méthodes sont traduisibles en calcul pour calculatrices ^élec-

troniques (« computer »)

^{et un} programme

spécial

^{a été élaboré. Une} méthode pour ^{le calcul} des lois dérivées pour ^les

représentations

de rotations sous Rn ~ R3 ^{a été mise au}

point.

Abstract. ²⁰¹⁴ Methods have been

developed

^for

analyzing

the reduction of the Kronecker

products

^for

orthogonal

^{and rotation} groups

using

^the S-function methods of D. E. Littlewood.

The

properties

^{of the}

spin representations

^{are discussed in some} detail and the method of difference characters used in the case of the even-dimensional rotation groups. These methods

are amenable to

computer

calculation and a

special

programme ^has been written. A method is

given

^for

calculating

^the

branching

^{rules for}

spin representations

^under Rn ~ R3.

JOURNAL ^DE PHYSIQUE ^TOME 30, AOUT-SEPTEMBRE 1969,

1. Introduction. ^- The

higher

^rotation groups,

Rn, play

^an

important

role in the group theoretical

descrip-

tion of atomic and nuclear wave functions and inter- actions

[1-4].

^The

properties

^{of the}

spin

representa- tions of the rotation _groups have been used in the

quasi-spin description

of atomic and nuclear shell

structure

[5-10].

^The

application

^{of the}

higher

^rota-

tion _groups,

especially R,

^and

R11,

^to

problems

ⁱⁿ

elementary particle physics

^is

of topical

^interest

[11-17].

The

analysis

of the reduction of the Kronecker

products

^{for the}

general

^{case of the}

spin representations

of the rotation _groups and in

particular

^{of the} repre- sentations of the even-dimensional rotation _groups

R2,

has received scant attention

[18-21].

The solution for the

special

^cases

of R4, R5 and R6 using

the known iso-

morphisms

^of

R4

with the double

binary

group, of

R5

with

SP4,

^and

of R6

^with

U4

is well-known

[2, 18-21].

Apart

^{from the}

special

difficulties attendant

spin representations

^{there are the}

primary

difficulties that

arise,

^{in the case} of even-dimensional rotation groups

R2.,

^from the fact that if the full

orthogonal

group

02v

is restricted to the _group

R2.

the character

(with X, =,4 0) separates

^{into two} irreducible _repre- sentations of

R2,1 [À1, À2, ..., Àv]

^and

[)’1’ X2, - - - - Àv] .

A

partial

^{solution to the}

problem

^of

reducing

^the

(1)

^Research

sponsored

ⁱⁿ part

by

^{the Air} Force Office of Scientific Research, Office of

Aerospace

Research, ^United States Air Force, ^under AFOSR Grant No. 1275-67.

Kronecker

products

^for

R2,

^{has been}

given by

^Little-

wood

[18] using

the method of difference charac-

ters

[18, 22-24].

^{In the} present paper ^we

complete

Littlewood’s treatment of difference characters to

give

a

general

method for the reduction of the Kronecker

products

^{of both true and}

spin representations

^{for the}

rotation _groups in an even,

R2v,

^or

odd, R2,+l,

^number

of dimensions.

Throughout

^we follow the notation of Littlewood

[18].

2. Kronecker

products

^for

02v

^and

°2V+l.

^{- The}

irreducible

representations

of the full

orthogonal

groups

°2V

^and

O2v+1 (the

groups

of orthogonal

^matrices

having determinant + 1)

may be labelled

by partitions

^of

integers

^{into not more than v} parts ^{and will be desi-}

gnated

^as

[À]’ == [À1, À2,

^...,

XJ’

^{where we attach a}

prime

^{to avoid} confusion with the

designation

^of

rotation _group

representations.

^The characters of

°2V

or

02v+i

may be

expanded

^as S-functions

using

Littlewood’s result

[18] :

where the sum is over all S-functions of the ^set

{ y }

^which

is

comprised

of all S-functions that arise for the

parti-

tions that are in the Frobenius notation of the form :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01969003008-9065500

i.e. the

partitions :

and p

^{is the}

partition weight.

An

S-function IXI

^may be written in terms of

orthogonal

group characters as :

where the sum is over all S-functions of the

set {8}

which involves all

partitions

^{into even}

parts only :

e.g.

{0}, {2}, {4}, {22}, {6}, {42}, {23}, {8},

^...

and the terms

arising

^{in the} S-function division are

taken as

orthogonal

group characters.

The results of

equations (1)

^and

(2) together

^with

the rule for the outer

multiplication

of S-functions and Littlewood’s method for

expressing

^{an S-function}

involving

> v

parts

^{as a} combination of S-functions

having v parts

allows the Kronecker

products

^of

the true

representations

^of any

orthogonal

group ^to be

readily

evaluated. Thus in the case of

04

^{we have :}

This method of

evaluating

^the reduction of the Kronecker

product

^{of the true}

representations

^{of the}

orthogonal

groups is amenable ^to machine calculation and is

incorporated

^{in a}

general

computer programme written

by

^{one of us}

(P.H.B.).

Extensive tables have been

published

^elsewhere

[2].

3. Kronecker

products

^of

spin representations

^of

02,

and

O2v+1·

^{- As well as the true}

representations

^of

the

orthogonal

groups there ^exist the so-called

spin representations.

^In

general

the Kronecker

product

^of

two

spin representations yields

^true

representations

while the Kronecker

product

^{of a}

spin representation

with a true

representation yields spin representations.

Associated with

°2’1 or O2v+ 1

^{there is a basic}

spin representation

^A

*

^{2 of}

^degree

^{2v. It is not}

difficult to show that

[18] :

for

O2v,

^{and :}

for

02V+ l’

where the asterisk

designates

the associated characters of the full

orthogonal

group. In the sum-

mation over the

orthogonal

group characters ^on the

right-hand-side

^of

equation (3 a)

the first summation is restricted to

integers

^{in the} range 0

2r v

^and the second to the _{range v} 2r 2v. Since in all the cases we shall consider the distinction between the

ordinary

characters and the associated characters is of

negligible significance

^we shall hence forth not

make _any discrimination.

Ignoring

this distinction leads to the S-function

expressions

^of

equations (3 b)

and

(3 d).

The

spin

characters for the other

spin representations

of the full

orthogonal

group may all be

expressed

^as

products

^{of the basic}

spin representation 2 ^witch

S-functions

by making

^the

expansion [18] :

where the sum is over all S-functions

involving

^self-

conjugate partitions

^of

weight p

^{and rank r.}

Thus in the case of

08,

^{we have :}

The

product

^{of a basic}

spin

character with an

S-function _may be

expressed

^{as a sum of}

spin

repre- sentation characters

by writing :

where E (n)

appears in the S-function

division {X}/{},

the summation

being

^over all S-functions

{}.

^Thus

in the case of

08,

^{we have :}

Equations (4)

^and

(5),

^{used in}

conjunction

^with

the results of

equations (3 a)

^and

(3 b), provide

^a

systematic

method for the reduction of the Kronecker

product

^of any

pair

^of

spin representations

^{or of a}

spin representation

^{with a true}

representation.

^The

results _may be checked

by

the usual dimensional methods.

4. Difference characters and true

representations

of

R2v.

^{- The}

representations [À1, À2,

^...,

Àv]

^of

02v

remain irreducible _upon restriction to the _group

R2v

except

^{in the case where}

Àv =1=

0 when the represen-

657

tation of

02v separates

^{into two} irreducible _represen- tations of

R2. :

Let us define the difference between the two

conju- gate

characters as :

Then we have :

and :

From first

principles :

The

product

of the difference characters

[1v]"

^times

an

S-function {À}

^will

yield

^a difference

character, simple

^or

compound,

^of

R2v.

^In

general

^{we find that}

the difference characters associated with the ^true

representations of R2v

^are

expressible

^as

[18] :

where the sum is over all S-functions of the

set I a I

which in Frobenius notation are of the form :

i.e. the

partitions :

{0}, {12}, {212}, {313}, { 23 }, {414}, {3221},

^...

and p

^{is the}

weight

^{of the}

partition,

^For

example,

in the case of

R8,

^we

readily

find that :

The

product

of the difference character

[1v]"

^with

an

S-function {À}

may be

expanded

^{as a series of}

difference characters

of R2v by writing [18] :

[1v]" {X} = 2[

⁺

¹¹

^{n2 +}

1, ..., 1) V

⁺

1]" (11)

where

E{n}

appears in

f X 1/{ p }

^{and the sum is over}

all S-functions of the

set {P}

which involves all

parti-

tions into an even number of parts ^of any

given magni-

tude. For

example

^for

Rs

^{we find :}

Equations (8 a), (8 b), (10)

^and

(11) together

contain sufficient information to make the reduction of the Kronecker

product

^of any

pair

^{of true} repre-

sentations

of R2v provided

^{we can} express the Kronecker square of the difference character

[1v]"

^{in terms of}

the characters of

R2,’

^{We first note that :}

a result that follows

immediately

^from

equations (6 a)

to

(8 b). By considering

^{each term on the}

right-hand

side

separately

^{we find} the somewhat

surprising

results that :

and :

where the

S-functions Xl( pL )

arise in the S-function

division {X}/{}

^{with the}

partition (À) defining

^the

principal

part ^{of the}

product (i.e. [2v]

^for equa- tion

(13 a)

^and

[2v-1]

^for

equation (13 b))

^{and the}

summation is

again

^{over the set of} S-functions which involves all

compatible partitions

^{of the}

type { 22r 12S }.

The terms that arise in the Kronecker

product

are the same as those

arising

ⁱⁿ

equation (13 a)

except that for _every

partition

^with

Àv i=

^{0 we}

replace X, by - Àv.

^{Thus for}

R8

^we

readily

find that :

and :

Let us now illustrate our

preceding

^remarks

by considering

the calculation of the Kronecker

product [2111]

^X

[211-1]

^for

R8.

^From

equations (8 a)

and

(8 b) :

Use of

equations (1)

^and

(2)

^{leads to} the result :

while

application of equations (10)

^and

(14)

^{leads to :}

from which we conclude that :

The reduction of

[2111]

^X

[2111] requires

^the

evaluation of the

product ^[2111]’

^X

[2111]" using equations (1), (2)

^and

(10)

^followed

by equation (11)

to

give :

from which we deduce that :

The result for

[211-1]

^X

[211-1]

will be the same as for

[2111]

^X

[2111] except

^{that for} every

partition

with

Àv =1= 0, X,

^is

replaced by - Àv

^{and vice versa.}

Again

^{we would}

point

^out that these results have been

incorporated

^{in our}

general

computer programme.

5. Difference characters and

spin representations

of

R2,.

^-

Equations (6)

^to

(8 b)

^are

equally

^{valid for}

the

spin representations

^of the rotation group

R2v.

The basic difference character becomes

[18] :

The difference characters associated with the other

spin representations

^of

R2v

may be

expressed

^as

[18] :

where the sum is over all

S-functions {E} ^involving self-conjugate partitions

^of

weight p

^{and rank r.}

It is not difficult to establish that the Kronecker square of the difference character is

expressible

^{in terms}

of S-functions as :

and that :

The establishment of these two

results, together

with the use of

equations (3 a)

^to

(8 b)

^{and of} equa- tions

(16)

^to

(18), gives

^a

systematic

method for

evaluating

the Kronecker

products

^of any

pair

^of

spin representations

^of

R2v.

As an

example,

^{let us consider} the evaluation of the

31 1 1 X 31 1 ¹

^{for R8}

Kronecker

product 2222 222 2

^for

Equations (6)

^to

(8 b)

^{lead to :}

From

equation (4)

^{we have :}

while use of

equations (16)

^and

(18) gives :

659

Combining

^{these two} results leads ^{to :}

Using

the result of

equation (18) readily

^{leads to}

the establishment of :

311 ¹

²

The terms

in 2 2 2 2

^{will be the same as above}

except

^{for the}

replacement ofÀv by - Àv

^when

Àv #-

^0.

6. Kronecker

products

^{of true and}

spin representa-

tions of

R2v.

^{- There is no}

difficulty

ⁱⁿ

extending

^the

preceding

^{results to calculate} the Kronecker

product

of ^a true

representation

^of

R2v

^{with a}

spin

represen- tation if the results :

and :

are noted

together

^with the relation :

where E{n}

appears

in {X}/{}

^{and the sum is over}

all

S-functions {}.

Using

^these

equations together

^with

equation (18) readily

^{leads to} the establishment of the

typical

^result

for

R8,

^{that :}

7. Resolution of the Kronecker _square for

02v

^and

02v +

1.

- In a number of

applications

of group

theory

in

physics,

^{it is} necessary ^to be able to resolve the terms

arising

^{in the reduction} of the Kronecker square of

a

representation

^{into its}

symmetric

^and

antisymmetric

terms. In the case of the true

representations

^{of the}

full

orthogonal

groups ^or of the odd-dimensional rotation _groups

R2, +

11 this

problem

may be

elegantly

solved

using

Littlewood’s methods of

plethysm [26].

In

general

^the

symmetric

^terms of the Kronecker square

[A]2

^are

just

those that arise in the

plethysm [A] 0 f 2}

while those of the

antisymmetric

^{terms arise}

in

[A] { 12 }.

We shall first consider the case of the

spin representations

^{of the full}

orthogonal

group.

From

equation (4)

^we find for the

symmetric

^{terms :}

and for the

antisymmetric

^{terms :}

The

plethysms involving

^the

S-functions {11}

^may

be evaluated

by

^{the usual}

procedures [18], [26], [27].

The

plethysms

for the basic

spin representations

^have

been considered

by

Littlewood

[18-20]

from first

prin-

ciples

^to

give

^for

O2v :

and for

and :

These two sets of

results,

combined with the results of

equations (22 a)

^and

(22 b),

allow any Kronecker square of the

spin representations

^of

O2v

^or

°2v+l to

be

separated

into their

symmetric

^and

antisymmetric

terms. Since the

representations

^of

°2v+l

^{are irre-}

ducible upon restriction of

02v+1 to R2,+l

^{there is no}

added

difficulty

ⁱⁿ

resolving

the Kronecker squares of the

spin representations

^of

R2v+1.

8. Resolution of the Kronecker

Squares

^for

R2v.

^-

As we saw earlier under the restriction

°2V

^-*

R2v

^the

irreducible

representations

^of

02v with X., =A

^{0 de-}

compose into two

conjugate representations

^of

R2v

^as

in

equation (6)

^{and to} evaluate the Kronecker _squares

we must make use of the method of difference charac-

ters. If

[À]

^{is a}

representation

^of

R2v

^{then we} may associate the

symmetric

^{terms of} the Kronecker square with the

plethysm :

or :

where the

+ sign

^{occurs if}

Àv

> 0 and the ^-

sign

if

Àv

^{0. The}

antisymmetric

^{terms will arise in the}

plethysm :

or :

The terms

arising

ⁱⁿ

M’@{2}

^and

[À]’ @ {12}

can be evaluated

using

the methods of the

preceding

section while those of

[A]’

^X

[A]"

follow from the methods

of section

4 for the case of true

representations

or section 5 for

spin representations.

^Thus

^only

^the

terms in

[XI " Of 2 1

^and

[X]II & f 12 1

^{need be consi-}

dered.

Noting equation (10),

^{we have in}

general

^{for the}

true

representations

^of

^{R2v :}

where we have used the usual result for

plethysms involving products

^and gp.,, ^{is the} coefficient of

{{JL}

ⁱⁿ

the inner

product f p I X ( m 1.

^{For the}

spin

^represen-

tations of

R2v,

^the

corresponding

^{result is :}

It should be

apparent

that from the ^two abov(

results that the

problem

of the resolution of the Kro- necker _squares for

R2v

^{would be}

complete

^{if the}

ple thysms

[1*1]"Of2}, [Jv]"01121, [(!)V]" @ ^{i2j

and

2 ^{@ { 12 }}

^{were known.}

To evaluate the

symmetric

^and

antisymmetric

terms in the Kronecker square of

2 v]

^{, let u.,}

first

put :

and :

We _may then write :

and :

Upon noting

^{the results of}

equations (3 a), (22)

and

(23) together

^{with the} fact that :

we

readily

^{find :}

and :

661

These two results used in

equation (30)

^{make it}

possible

^to resolve the Kronecker _square of the

spin representations

^of

R2v

^into

symmetric

^and

antisym-

metric ^terms.

The evaluation of the

plethysms [lv]" & 112 1

^and

[1v]" ø {2}

^{is a somewhat more} difficult task. We

proceed by

^first

noting

^{that :}

and :

Furthermore :

and :

The terms in

[1v] [1 v-I

^-

1]

^follow

directly

^from

equation (13 b)

^{and we} have for v even :

and :

while for v odd :

and :

Evaluation of the

plethysms

^of

equations (38 a-d),

and use

of equation (28) readily

^{leads to the evaluation}

of the

plethysms involving [1v]

^and

[1 v-I

^-

1]

ⁱⁿ

equations (36 a)

^and

(36 b).

^{Thus in the case of}

Rg,

we have :

and since :

Using

^{these results in}

equations (36 a)

^and

(36 b)

leads

directly

^{to the results :}

and :

9.

Branching

rules for

spin representations

^under

Rn -+ R3.

^-

Armstrong

^and

Judd [28]

^have

recently

shown that the

spin representations

of the rotation groups

R,, play a

fundamental role in the

analysis

^of

the structure of electron

configurations.

^{The bran-}

ching

rules for the

decomposition

^{of the}

spin

^characters

of Rn

into characters

of R3

^are

especially important

ⁱⁿ

their treatment.

The

representations

^of

R3 depend

^{on a}

single

para- metric

angle

⁰ and the character associated with the

representation

^{DEJ3 is :}

where j

may be

integral

^or

half-integral

^and p ^runs

through positive integers.

^The

representations

^of

R2v

and

R2v+1

each involve v

parametric angles 6g

^{and in}

particular

the character

[1]

^{for the vector} represen- tation rlll

of R2v

is of the form :

while for

R2v+1

^the

corresponding

character is :

Under the restriction

Rn --+ R3,

^{the v}

parametric angles

become related and it is then

possible

^to express the results of

equations (40)

^and

(41)

^{in terms of a}

single parametric angle

^{0. If under}

Rn --+ R3,

^we

have :

then

comparison

^of

equations (40)

^and

(41)

^with

equation (42)

allows the

relationships

between the

v

parametric angles

^{to be fixed}

immediately.

For

example,

^{if under}

R2l+1

^-*

Rs,

^{we have}

[1]

--->

[l]

then

comparison

^of

equation (41)

^with

equation (39)

shows that the

required relationship

^{is :}

Thus if

under R9 - R3

^{we have}

[1]

^-+

[4]

^then

we must take :

If, however,

^under

R9 R3

^{we have}

[1]

->

[0]

+

[1]

+

[2]

^then

comparison

^of

equation (41)

^with

equation (42)

^{leads to} the choice :

Likewise,

^{if under}

R6 -* R3

^{we have}

[1]

^-*

[0]

+

[2],

^then

comparison

^of

equation (40)

^with equa- tion

(42) gives

the relations :

Having

^{defined the}

decomposition

^{of the vector}

representations

under the restriction

R. -->- R3,

^{as in}

equation (42)

^we may

immediately

^{find the corres-}

ponding decomposition

^for any other true represen- tation

[À]

^of

Rn by evaluating

^{the terms in the}

ple- thysm [21] :

The character associated with the basic

spin

^cha-

racter A of

R2’)+1

^{may be}

expressed

^{in terms of v} para- metric

angles 6e;

^to

give :

where the summation is over all

possible

combinations of the

plus

^{and minus}

signs.

^{In the case of the even-}

dimensional rotation _group

R2v,

^{we have :}

where the summation is over all combinations of

signs involving

^{an even} number of minus

signs.

^The

conju-

gate

spin

^character

A,

^{is of the same form}

except

^that the summation is over all

possible

combinations of

signs involving

^an odd number of minus

signs.

If the

decomposition

of the character of the vector

representation

^FE’3

of R2v

^or

R2v+1

under the restriction

Rn R3

^is defined then the

relationships

among the v

parametric angles

is fixed and _may be used in

equations (48)

^and

(49)

^{as the case} may be.

Compa-

rison with

equation (39)

^leads

immediately

^{to the}

decomposition

^{rule for the}

spin

^character.

Thus if under

R9 R3

^{we have}

[1]

^--*

[4]

^the

relationships

between the four

parametric angle

^are

fixed

by equation (44)

which when

employed

ⁱⁿ

equation (48),

^{and the result}

compared

with equa- tion

(39),

^leads

immediately

^{to the result :}

which may be verified ^as

dimensionally

^correct.

If, however,

^under

R9 -¿. R3

^{we have}

[1]

-->

[0]

⁺

[1]

+

[2]

^{then we must use the}

parametric angle

^relation-

ships

^of

equation (45) giving :

Likewise,

^{if under}

R6 -+ R3

^{we have}

[1]

-->

[0]

+

[2]

then use of

equation (46)

ⁱⁿ

equation (49) gives

^the

result :

The

corresponding

result for

A2

^{is identical to that}

for

Al. Comparison

^of

equations (48)

^and

(49)

^with

equations (40)

^and

(41)

^leads

immediately

^{to the result}

that if under

R2l+1

^-*

R3

^{we have}

[1]

^--*

[1]

^and

A -*

Eg, ,[L]

while under

R2l+2

-->-

R3

^{we have}

[1] -+ [0]

⁺

[1]

^{then also}

A, -->- EgL[L].

The

decomposition

of the basic

spin representations

under

R2l+1

--->-

R3

^when

[1]

-->-

[l]

^is

given

^{for I == 2}

to 10 in table I and some

typical decompositions

^for

the basic

spin representation

^for

R2.

-->-

R3

in table II.

10. Selection rules and the _group

R2,.

^{- It is well-}

known that if the

bra A

^and

ket B >

^{of a matrix}

element A I H I B >

^transform

according

^{to the}

irreducible

representations rA

^and

r B

^{of a} group G and the

operator

^H

according

^{to the}

representation rH

of G then the

vanishing

of the matrix element is assumed unless :

In the case of the _groups

R2v+1, R4v

^and

O2v,

^the repre- sentations ^are all

self-adjoint

^{and hence}

ri

⁼

rH.

However,

^for

R2v

^where

X, =A

^{0 and v}

odd,

^{we have :}

Tables of the numbers

G’(rA rB rH),

i.e. the number of times

rH

^{occurs in}

rA

^X

rB,

^{for the}

group R6

^have

been

given

^earlier

[21].

^In

using

these tables for

determining

^selection

rules,

it is essential to note equa- tions

(53)

^and

(54).

Thus the selection rules for the matrix elements of ^an operator

transforming

^as

[222]

of

R6

^will be found in the table of

G(rA rB[22-2])

and not

c(rA rB[222]).

Similarly,

^{if we have on} operator H(rHYH) ^that transforms as the

rH representation

^{of a} group ^G and

as the _yH

representation

^{of a}

subgroup g

^{of G then}

the matrix element :

will

certainly ^vanish,

unless either

[26] :

or :

11. Conclusion. ^- We have

attempted

^{to demons-}

trate in this paper the methods

by

which the Kronecker

products

^{of the}

spin

^{and true}

representations

^{of the}

orthogonal

and rotation groups may be calculated in

an

unambiguous

and self-consistent manner. The

problem

^of

resolving

the Kronecker _square into its

symmetric

^and

antisymmetric

^terms has been solved

using

Littlewood’s methods of

plethysm. Finally,

the

decomposition

^{of the}

spin

characters of

Rn

^into

663

TABLE I

DECOMPOSITION OF THE BASIC SPIN REPRESENTATION UNDER .

TABLE II

DECOMPOSITIONS OF BASIC SPIN REPRESENTATIONS UNDER

those of

R3

has been shown to be

readily

^amenable

to calculation.

The elaboration of the above methods has made it

possible

^{for one of us}

(P.H.B.)

^to

complete

^a computer programme which ^not

only

handles the character

theory

^{of the true}

representations

^{of the}

unitary,

^ortho-

gonal

^and

symplectic

groups but also the

spin

^character

theory

of the rotation _groups. This _programme is

being

^used

extensively

^{in a number of}

applications

in atomic

physics

and the authors will be

pleased

^to

consider

specific requests

for extensions to

existing

tabulations of group character

properties.

Acknowledgement.

^{- We are}

grateful

^to

Mr

J.

^G.

Cleary

^for

making

^the

computations

^for

tables I and II on the

University

^of

Canterbury’s

IBM

360/44 computer.

REFERENCES

[1] JUDD (B. R.), Operator Techniques

^{in Atomic}

Spec- troscopy, 1963,

McGraw-Hill Book

Company,

^Inc.,

New York.

[2]

^WYBOURNE

(B. G.), Symmetry Principles

^{and Ato-}

mic

Spectroscopy,

^1970,

John Wiley

^{and Sons,} Inc., ^{New York.}

[3] JAHN (H. A.),

^Proc. Roy. ^Soc.

(London), 1950,

^{A 201,}

516.

[4]

^ELLIOTT

(J. P.),

^Proc. Roy. ^Soc.

(London),

1958, A 245, ^128.

[5] JUDD (B. R.),

^Second

Quantization

^{in Atomic}

Spec- troscopy,

^{1967, The}

Johns Hopkins

Press, ^Balti-

more,

Maryland.

[6]

FLOWERS

(B. H.)

^and SZPIKOWSKI

(S.),

^Proc. Phys.

Soc.

(London),

1964, 84, 193.

[7]

^FLOWERS

(B. H.)

^and SZPIKOWSKI

(S.),

^Proc. Phys.

Soc.

(London),

1964, 84, 673.

[8]

^HELMERS

(K.),

^Nuclear

Phys.,

^{1965, 69, 593.}

[9]

^HECHT

(K. T.), Phys.

^{Rev., 1965, 139, 794.}

[10]

SZPIKOWSKI

(S.),

^Acta

Phys.

Pol., 1966, 29, 853.

[11]

^GURSEY

(F.),

^Annals of Physics, 1961, 12, ^91.

[12]

^NE’EMAN

(Y.), Phys.

Letters, 1963, 4, ^81.

[13]

^PEASLEE

(D. C.), J.

^Math.

Phys.,

1963, 4, 910.

[14]

^COCHO

(G.),

Phys. Rev., 1965, 137, ^B 1255.

[15]

JOSEPH

(D. W.), Phys.

Rev., 1965, 139, ^B 1406.

[16] JOSEPH

^(D.

W.)

^{and SMALLEY}

(L. L.), Phys.

Rev., 1966, 150, 1209.

[17]

^NE’EMAN

(Y.)

^{and OZSVATH}

(I.),

Phys. Rev., 1965, 138, ^{B 1474.}

[18]

LITTLEWOOD (D.

E.),

^The

Theory

^of

Group

^Charac- ters, ^2nd ed., 1950, ^Oxford

University

Press, London.

[19]

LITTLEWOOD

(D. E.),

Proc. Lond. Math. Soc., 1947, 49

(2),

^307.

[20]

LITTLEWOOD

(D. E.),

Proc. Lond. Math. Soc., 1948, 50

(2),

^349.

[21]

^WYBOURNE

(B. G.)

^{and BUTLER}

(P. H.), J. Physique,

1969, 30, ^181.

[22]

^MURNAGHAN

(F. D.),

^The

Theory

^of

Group Repre-

sentations, 1938, ^The

Johns Hopkins

Press, ^Bal- timore,

Maryland.

[23]

^MURNAGHAN

(F. D.),

^The

Unitary

and Rotation

Groups,

^1962,

Spartan

Books,

Washington,

^D.C.

[24]

^NEWELL

(M. J.),

^Proc.

Roy.

^Irish Acad., 1951, A 54, ^143.

[25]

^BUTLER

(P. H.)

^{and WYBOURNE}

(B. G.), J.

Phy-

sique,

1969, 30, ^181.

[26]

^SMITH

(P. R.)

and WYBOURNE

(B. G.), J.

^Math.

Phys., 1967, 8, 2434.

[27]

^SMITH

(P. R.)

and WYBOURNE

(B. G.), J.

^Math.

Phys., 1968, 9, 1040.

[28]

^ARMSTRONG

(L.)

^and JUDD

(B. R.),

^Proc. Roy. ^Soc.

(London),

ⁱⁿ press.