Applications of s-functional analysis to continuous groups in physics

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Applications of s-functional analysis to continuous groups in physics

P.H. Butler, B.G. Wybourne

To cite this version:

P.H. Butler, B.G. Wybourne. Applications of s-functional analysis to continuous groups in physics.

Journal de Physique, 1969, 30 (10), pp.795-802. �10.1051/jphys:019690030010079500�. �jpa-00206842�

APPLICATIONS

OF S-FUNCTIONAL

ANALYSIS

TO

CONTINUOUS

GROUPS IN

PHYSICS (1)

By

^{P. H.} BUTLER and B. G.

WYBOURNE,

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

(Reçu

^{le 14 mars}

1969.)

Résumé. 2014 Les fonctions S, ^telles

qu’elles

^{ont été}

développées

par Littlewood, sont

passées

en revue dans le but de

simplifier l’algèbre

des groupes ^continus. La division de la fonction S est définie et la théorie a été

développée jusqu’au point

^{où un} programme pour calculatrice

électronique (computer)

^{a été établi, ce}

qui

permet ^le calcul des

produits

Kronecker, des lois dérivées, ^des

pléthysmes

^{sur deux} variables, ^{et des}

produits

internes du groupe

symétrique.

Abstract. ²⁰¹⁴ S-functions, ^as

developed by

Littlewood, ^are reviewed with the aim of

simpli- fying

^the

algebra

of continuous groups. S-function division is defined ^{and the}

theory developed

to a stage ^{where a}

computer

programme ^has been written that

performs

^Kronecker

products, branching

rules,

plethysms

^{on two} variables, ^{and inner}

products

^{of the}

symmetric

group.

1. Introduction. ^- In the past

decade,

theoretical

physicists

have shown an

unprecedented

interest in the

theory

of continuous _groups and its

application

to a wide _range of

physical problems. Notable,

among the _many

applications,

^{has been the use of}

the

compact

continuous groups ^to describe the _sym- metry transformation

properties

^of

N-particle

^atomic

and nuclear wave functions

following

^the

early

^work

of Racah

[1, 2]. Physicists

have tended to concen- trate

primarily

^{on the}

development

of continuous groups, in the tradition of Elie Cartan

[3]

^and

Sophus

Lie

[4] by considering

^the

properties

^of infinitesimal transformations. Hermann

Weyl’s

^{book on "The}

Classical

Groups" ^[5]

^has

undoubtedly

^{exercised a}

considerable influence in these

developments.

An alternative

approach

^{to the}

theory

^{of continuous}

groups, which

complements

^the earlier work of Cartan and

Lie,

^{has been}

developed by

^{D. E.} Littlewood ^as

a natural consequence of Schur’s

original

^thesis

[6]

on the

properties

^{of invariant} matrices. Littlewood’s

treatment circumvents the

study

of infinitesimal trans-

formations

by considering

^the

properties

^of

special

functions of the ^roots of the matrices that characterize the elements of the continuous _groups. This

approach

obviates the need to obtain the group characters

explicitly.

^These

functions,

^{known as} Schur-func-

tions,

^or

simply

^as

S-functions,

have been used

by

Littlewood to find

relatively simple

^formulae

relating (1)

^Research

sponsored

ⁱⁿ

part by

^{the Air Force}

Office of Scientific Research, ^{Office of}

Aerospace

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

the characters of

representations

^{of the}

unitary,

sym-

plectic, orthogonal

and rotation _groups.

Methods for

calculating

^outer

products

^{of S-func-}

tions are well known

[7]

^{and these are}

developed

^so

as to

give simply

^and

unambiguously

the Kronecker

products

for all the above groups. A method for

determining

^{the inner}

product

of S-functions without the usual recourse the character

tables,

^{is used}

together

with a more recent

development,

^{that of}

plethysm,

^to

give

^{us a}

general

^{method of}

uniquely determining branching

rules between the above _groups and their

subgroups.

This _paper describes the relevant

theory

^{and how}

it is used to

give

^{a set of} computer programmes ^to

perform

^the

complete algebra.

2. S-Functions and Continuous

Groups.

^{- In this}

section,

^{we shall}

briefly

outline the mathematical concepts ^{and notations}

required

^{in the}

subsequent development

^{of this} paper.

Partitions. ^- A set of r

positive integers

^{whose sum}

is n is said to form a

partition

^{of n.} An ordered

parti-

tion is one where the

integers

^{are ordered from}

largest

to smallest

(or

^vice

versa).

^All

partitions henceforth,

will be so ordered with the notation that ^a Greek letter will denote ^a

general partition

^{so :}

(X)

^or

(À1, À2,

^...,

x,).,

and this will be assumed ^to be a

partition

^{of n. A}

Latin letter will denote ^a

partition

^{into one}

part only.

A

Young diagram

^is associated with each

partition.

It is the

graph

^{of r}

rows, Ài

^dots

(or squares)

^{in the}

i th _row, with each row left

justified.

^A

conjugate

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019690030010079500

796

partition

is formed

by interchanging

^{the rows and}

columns of the

graph,

and will be denoted

by (À) :

Some

partitions

^are

self-conjugates

e.g.

(332).

Partitions with

repeated

parts ^are often written with

a

superscript denoting

the number of times the part

occurs, thus

(33221)

⁼

(32 221 ) .

Frobenius’ notation is sometimes of use. The

leading diagonal

^{of the}

Young diagram

^{is defined to}

be the one that starts in the top ^{left hand corner. For} each dot on the

diagonal

^we write down the number of dots to the

right

^of

it,

^{and below}

this,

^{the number}

Symmetric

Functions. ^- A

symmetric

^{function on}

k variables _ai is one that is

unchanged by

any permu- tation of the variables. Two such functions ^are of interest in this _paper.

1)

Mononomial

Symmetric

Functions. ^- If

(p)

^{is a}

partition,

^we define the mononomial

Sp

such that :

where the summation is over all different

permutations

of the a’ s. For

example,

^{if k = 3 :}

2) Homogeneous

^Product

Sums, hn.

^{- The}

homogeneous product sum hn

is defined to be the sum over all of the mononomials

Sp,

^p

^being

^a

^partition

^{of n :}

S-functions.

^{- If}

CA)

^{is a}

partition

^{of n, the S-function}

(x)

^{is the} determinant of the

h¡ s

^{defined as follows :}

the s and t

being subscripts

^{for the row and column}

respectively.

We extend the definition of the homo- geneous

product

^{sums to}

include ho

^{= 1}

and h.

^{= 0}

Littlewood has shown how to express any S-function in ^terms of mononomials

by appropriately labelling

the

Young diagram corresponding

^to

partition (X) [8].

The

Symmetric Group.

^{- Under the}

operations

^of

the _group, the

symmetric group Sn

with n ! elements is

split

into classes _p with

hp

elements. Each class is the

complete

^{set of}

conjugates

^{of a}

given

^element.

The irreducible

representations

^{of the} group may be

placed

^{into a one to one}

correspondence

^{with the}

partitions

^{on n. Because}

conjugate

matrices have the

same characteristic

(i.e.

spur ^or

trace)

there exists a

unique

^number

xP(03BB),

the characteristic for a

particular

class of ^a

representation.

^{The set} of characteristics of a

representation

^{is known as} the character of the

representation.

^{If we define a function :}

we may prove that it is in

complete correspondence

with the

representations

^{of the}

symmetric

group.

Littlewood

[9]

has shown that this definition is

entirely equivalent

^to the definition of the S-function

given ^earlier,

^to

give

^{us a}

complete isomorphism

^bet-

ween

operations

^on S-functions and

operations

^on

representations

^{of the}

symmetric

groups. The corres-

pondence

between the

symmetric

groups and the full linear _group

paved

^the way for Littlewood to express the

algebra

of the continuous groups in ^terms of S-functions. This result is of

key importance

^{in the}

following

^work.

3. Dimensions of

Representations

^of

Groups.

^{- Di-}

mensions of

representations

^{are used}

extensively

^to

check all

branching rules, products

^and

plethysms.

Several authors

[10] give

formulae for the dimension

(degree) f{À}

^{of a}

representation {X}.

Symmetric Group S n :

where {À}

^{is a}

partition

of n into ^r

parts.

^{A for-} mula more suited ^to hand calculations is

give . by

Robinson

[11].

The

Unitary Group

The

Symplectic Group,

^>

The

Orthogonal

and Rotation

Groups On, R. : a)

For odd dimensions n ⁼ 2v + ^{1 :}

b)

^{For even dimensions}

except ^{that for the}

orthogonal

group in the case of

X, =,4

0 when the dimension is twice the above.

The

Exceptional Group G2 :

4. Outer

Multiplication

^of S-functions. ^- The

product

^{of two} S-functions on different sets of variables

-

corresponding

^{to the}

product

^of

representations

^of

different

symmetric

groups ^- is known as the outer or

ordinary multiplication

of S-functions. It is of

key importance

^{in the}

algebra.

The rules for

performing

the

operation using

^the

Young diagram representation

have been

given

many times

[7] although,

^{so far as}

it is

known,

^{it has not}

suggested

^{how to}

systematically

cover all

possibilities

^{for the}

graphs.

The S-functions

appearing

^{in the}

product :

are those which can be built

by adding

^{to the}

graph of {À} f1-1 symbols

^a, f1-2

symbols P,

03BC3

symbols

y, etc., in this order and in the _ways

specified by

^the

following : 1)

^{No two identical}

symbols

appear in the ^same column of the

graph.

2)

^{If we count the}

^{oc’ s,} 3’ s, y’ s,

etc., from

right

to

left, starting

^{at the} top, ^{then at} all times while the

count is

being ^made,

the number of a’ s ^must be not

less than the number of

P’s

^{which must not be less}

than the number of

y’s,

^{and so on.}

3)

^The

graphs

^we obtain after the addition of each

symbol

^{must be}

regular,

i. e. the

corresponding parti-

tion must be ordered.

The

principal

part ^{of the}

product

^{is the term obtai-}

ned when the

partitions

^are

simply added,

^{i.e. the}

partition {Àl + [03BC1’ À2 + [1.2’ ..., Ài + [1.i, ...}.

^It

corresponds graphically

^to

putting

all the (x’ s in the first _row,

the P’s

in the second and so on. The other

terms in the

product

may be

systematically produced by removing

the last element and

trying

^{it on the next}

lower

line,

^then the next, ^etc. When it will fit nowhere else remove the second to last element also.

Try

^to

fit this in the same

fashion,

^{if no}

place

^{is found remove}

the third to last

element,

^{when one is found to} fit

try

to

replace

the elements in their

highest positions.

^Note

also

that,

^for

example,

for the third _y we may

place

^it

on the same

line,

^or

below,

the second _y but not above it.

This

operation

is checked

dimensionally by

^{use of}

the

equation :

where

(X)

^{is a}

partition of n,

^and

(03BC)

^{of m.}

For

example {31} {21} :

note that the

graphs :

all break the rules

given

^above.

Thus we obtain the

expansion :

798

5. S-function Division. ^-

Frequently

^the

algebra requires

the evaluation of the sum of

S-functions {v}

which when

multiplied by

^a

particular S-function {03BC}

give

^a

particular S-function {À},

the coefficient

of {v}

being

the coefficient

of {À }in

^{the outer}

product.

^Hence

we define the

(outer)

division of

S-functions {À} /{03BC}

to be :

where

r 03BCv03BB

^{is the same as} the coefficient in the ^outer

product :

(’1)

The evaluation of the division is somewhat easier than each

product,

^thus

considerably simplifying

^the

calculation.

We have the

graph of {03BC}

^{and wish to know all}

possible

ways of

adding

^{elements to form the}

graph of {À} given

the rules for the

product.

^{To evaluate}

the

division,

^{draw the}

graph

^for

(h)

^{with squares}

instead of

dots,

^and fill up the left hand top ^{corner with} the

graph corresponding to {03BC}. Graph {03BC}

^{must fit}

entirely inside {À}

^or the result must be null. The

remaining

squares ^are then labelled

by a’s, 3’s, Y’ s,

etc.,

by

rows,

starting

^{in the} top

left,

^as

given by

rules 1 and 2 of the

product

and also with :

3)

^The

symbols

^{must not} decrease when

reading

left ^to

right

^{across a row,} i.e. there ^{must not} be an a to the

right

^{of a}

03B2,

^etc.

4)

The resultant S-function must be ordered.

For

example {4211}/{211} :

note that the

graphs :

are not allowed

by

the rules.

S-function division _may also be

performed by

^use

of the

properties

of isobaric determinantal forms

[8] : where 03BBs,

is the s-th part ^{of the}

partition (03BB) and 03BCt,

^is

the t-th part ^{of the}

partition (03BC)

^and thus s labels the

rows and t the columns of the determinant. In the

case of the

preceding example,

^{we have :}

as before. In

practice,

^the

preceding

^{method is to}

be

prefered

^for machine calculation. The nota-

tion {À} / {fL}

^{is to be}

prefered over {À/fL}

^{since we}

may

easily

^{show :}

6.

Expression

^{of an} S-function in

Symmetric

^Parts.

- When an S-function is defined from character

theory

^the

problem

^of

expressing

^{it in terms of sums}

and differences of

symmetric

^{functions is a} little diffi- cult.

However,

^the

equivalence

between that defini- tion and ^our definition renders the

problem

^trivial

since our

defining

determinant is in ^terms of the

hr s (eq. 3) :

Sometimes it is useful to know the

expansion of {À }

in terms of

products

^{of the}

antisymmetric

represen-

tations {1 r}.

^Because

hr == {1r},

^we may take

conju-

gates of the above relation.

Alternatively

^{we define}

the

elementary symmetric function ar

^{as the monono-}

mial

corresponding

^{to the}

partition ( 1 r) :

^:

leading

^to the result

[9] :

7. Inner

Multiplication

^of s-functions. ^- The eva-

luation of the inner

product {À} 0 { tL}

^is

considerably

more

complex

^{than the outer}

product.

^Numerous

attempts

have been made to

simplify

^the

problem,

with

varying degrees

^{of success. Most of these}

attempts

have used the character table for the _group in

question,

however Littlewood

[12]

^has

developed

a much ^more flexible method where the character

tables are not

required.

^The

key

^{theorem is stated}

without

proof :

where

r p03C303BD

has the usual

meaning

^{for the outer}

product.

The

symmetric representation

^{is the}

identity

^element

for this

operation :

(X)

^a

partition

^{on n. Inner}

multiplication

is distri- butive with respect ^to

addition,

^{hence all that is}

required

for the evaluation of _any inner

product

^{is for one}

S-function to be

expressed

^{in terms of}

symmetric parts,

and the

appropriate

^outer

multiplications performed.

To evaluate :

the steps ^{are :}

where

(vi)

^{is a}

partition

^of

^i,

^{the sum is over all such}

partitions,

^and

rVavb.Vc03BC

^is the coefficient

of 03BC

ⁱⁿ

the outer

product {va} { Vb}, ..., { Vc}.

We _may check ^our result

by noting [9] :

Various relations _among the coefficients are of

importance [9] :

The

conjugate

^{relations are of}

importance

ⁱⁿ

listing

tables of inner

products, reducing

the number

by

^a

factor of four. Relations

(21 b)

express ^not

only

^that

the

operation

^is

Abelian,

but that the coefficient

of {v }

in the

product {À} 0 {03BC} equals

the coefficient

of {03BC}

in the

product I X I o {v 1.

^{This means that if we define}

inner division

analogous

^{to outer}

division,

^we

merely

have the

operation

^{of inner}

multiplication :

Trivial consequences of this that are often relevant

to

problems

^{are that the}

identity I n I

is contained

only

in

products

^{of the} type

{À}o{À},

^{and the}

antisymme-

tric

representation {1 n}

^is

only

contained in those of the

type {X}o{X}.

8. Characters of

Subgroups

^as S-functions. ^- The

set of all

non-singular

matrices of order n2 form a

group, the

general

^linear group

GL..

The subset of all

unitary

matrices also forms ^a _group, the

unitary

group

Un.

^The compact

representations

^{of these two}

groups have the same characters and Littlewood has shown

[8], [9]

that the S-functions on n variables are

the

simple

characters of the _groups.

The full

orthogonal,

rotation and

symplectic

^groups

of

degree n

^{all occur as}

subgroups

^of

Un.

^Little-

wood

[13]

^has

expressed

the characters of the ortho-

gonal

^and

symplectic

^{groups in terms of} S-functions :

where

(y)

^and

(8)

^are

partitions of p

^{and occur in the}

Frobenius series :

The character

theory

for the rotation _groups are essen-

tially

^{the same as for the}

orthogonal

groups except ^when the _group dimension is even

(n

⁼

2v)

^and

X, 0

^0.

In most of these cases, it is necessary to resort to the method of difference characters

[8] though

^{in the}

particular

^{cases of the} group

R,

^and

R6

^{it is}

possible

to use

simpler

^{methods as} will be discussed later.

The

exceptional

group

G2

occurs as a

subgroup

of

R7

^{and is}

important

in the classification of the

states of electrons ^or nucleons in

equivalent

^orbitals.

The character

theory

^of

G2

is discussed in ^a later section.

9. Reduction of the Number of Parts of an s-function.

- Under the

operations

of the restricted _groups an

S-function defined on n

variables,

where n = 2v ^or

n = 2v +

1,

^and

having

^{more than v} parts ^is

equiva-

lent to a series of S-functions on the same n variables but not

having

^{more than v}

parts [13].

The S-function is

expressed

in the form :

Ignoring

^a

possible change

^of

sign

^{for some transfor-}

mations,

this S-function is

independent

of r and will be

denoted {À: 03BC}.

^{It is}

expanded

^to

give

^{a series of}

S-functions

using

the relation :

where the sum is over all

S-functions {a}, ^being parti-

tions

of p.

This relation is used as often as necessary to reduce all terms to those of no more than v

parts.

Two

special

^{cases of this}

equivalence

^{relation are}

often useful. In n variables for

unitary

transforma-

tions,

^{we have :}

Ignoring

^the

change

^of

sign

^{when n = 2v}

only

^for

transformations of

negative determinant,

^we have also : This latter relation

gives

the well known

particle-hole

correspondence.

800

10.

Branching

^{Rules. -} Under restriction ^to a

subgroup,

the characters of a group

decompose

^into

a sum of characters of the

subgroup [13].

^{For the}

unitary

group in ⁿ

variables,

the characters are expres- sed as S-functions of _up to n

parts

^but for the restricted groups ^{on n}

variables,

^into

only v

parts. ^{Prior use of} relation

(26)

^{allows us to use the}

following

^relations

without

producing

non-standard

symbols.

For the

Orthogonal Group :

where the sum is over all S-functions of even parts

only :

and where the terms of the division are taken as

orthogonal

group

characters,

^{i.e. :}

The

Symplectic Group :

^{For the}

symplectic

group the result is the same as for _eq.

(31 ) apart

^{from the}

repla-

The Rotation

Group :

^{For odd} dimensions the charac-

ters are the same as for the

orthogonal

group but for

even dimensioned _groups the characters

with [03BCv

^{#- 0}

decompose

^{into two}

conjugate

characters :

The

Exceptional Group G2 :

^The group

G2

^{is a} proper

subgroup

^{of the seven} dimensional rotation _group and

Judd [10]

has derived the

branching

^rules

by using

the infinitesimal

operator approach,

^to

yield

the result :

where the sum is over all

integral

^values

of i, j, k

^satis-

fying

the relations :

is used to remove characters which do not

give regular representations

^of

Gz.

11. Kronecker Products for the Continuous

Groups.

- Since we may express the characters of the

unitary, symplectic

^and

orthogonal

groups in ^terms of S-func-

tions,

the reduction of the Kronecker

products

^{of these}

groups may be done in terms of the outer

product

of S-functions and then

performing

^the

appropriate branching

^to get back the characters of the _group.

Kronecker

products

^for

G2

^are done in the same manner

by

^{a two stage} process,

expressing

the charac-

ters

of G2

^{in terms of those}

of R7,

thence into S-functions and so on. The

expression

^{of the} characters of

G2

in ^terms of those of

R7

^is

performed by noting

^{that in}

the reduction

R7 --> G, [u1 u2 0]

^contains

(UIU2)

^as

the term

of highest weight.

^{The terms of lower}

weights

may be

systematically

^removed

by

subtraction.

For

example,

^{in the case of}

(21),

^we may derive :

may be

expressed

in S-functions to

give :

For even dimensional rotation _groups,

products

ⁱⁿ

only

^two groups have been

separated satisfactorily,

the groups in four and six

[14]

dimensions. In six

dimensions,

Littlewood

[15]

has shown that the _group is

isomorphic

with the four dimensional full linear group and the

correspondences :

may be

established,

^thus

allowing

^{us to}

perform

^the

products easily.

^{For the} four dimensional rotation group there is a 2:1

homomorphism

with the double

binary

^full linear group and the

correspondences :

may be made.

We _may

readily

deduce that the

separation

^{of the}

Kronecker

product

ⁱⁿ

R4

^is

given by :

12.

Plethysms

for

GL2

^and

R3.

^-

Plethysms

^o

characters on two basis variables are often of use in

physics. Plethysms

^for

GL2

may be evaluated

by restricting

the results for the same

plethysm

^{on an}

unrestricted number of variables. In the absence of such a

table,

^we may generate ^the

plethysm required by

^use of the recursive relation

[16] :

The

following polynomial expansion

^{due to Little-}

wood

[17],

^{for the}

plethysm

^{of a}

R3 representation

and an S-function into

only

^one

part,

^{is more suited}

to machine calculation :

where

Kr

is the coefficient of

P-r

^{in the}

expansion

^{of :}

This

expression

holds also for

spin representations.

A similar

expression

^{for the}

plethysm

^{of a}

GL2

repre-

sentation allows us to make the

following isomorphism

between

GL2

^and

R3 :

13.

Branching

^{Rules to}

R3’

^{-r The use of the}

algebra of plethysm [9], [14], [8]

^{allows us to}

calculate, directly

and

unambiguously,

^the

branching

rules between _any compact group and a

subgroup

^of lower dimension or a direct

product

^{of such} groups

[19], [20]. Quite generally, given

^{that the} unary character

[1]

^{of the}

larger

group

decomposes

^{to a sum} of characters ^A under the

restriction,

^then any character X will decom- pose into the ^sum of characters

given by

^{A 0 X.}

The

branching

^{rules for} any group

R3

may therefore be

easily

calculated after

defining

^our unary de-

composition.

For

example,

^to calculate the

branching

^of

[210]

of

R7

^to

R3

^{we note that}

[100]

^of

R7

^{branches to a}

single

^F state, i.e. the

representation [3] of R3.

^Hence

the

decomposition

^of

[210]

^is

given by [3]

⁰

[210].

We _express

[210]

^{in terms of}

S-functions,

^{and then these}

into sums and

products

^of

partitions

^{into one} part,

giving :

Using

the result

[9]

^{that the}

plethysm

is distributive

on the

right

^with

respect

^to both addition and multi-

plication, by using equation (38),

^and

by multiplying together

^the

R3 representations

in the usual _manner,

we obtain :

This method of

performing

^the

plethysm

^{on the}

symmetric parts

^{of the}

right

hand side is not

always

the easiest method. Often it is easier to make use

of the relation

[16] :

by expanding

the S-functions in terms of their anti-

symmetric

parts.

The

saving

in labour for

classifying

the orbital states

of maximum

multiplicity

^is immense. For

example,

in the t shell ^we use

representations of R29.

^The unary

decomposition

^is

[100,

^...,

0]

->

[14] = { 28 }.

^Thus

the states of maximum

multiplicity

^{for the} quarter filled shell labelled

by [114]

^are

easily

^{found to be}

just

the terms in :

This has been

expanded using

^the

computer

^{and there}

are found ^to be

34,670

^{states of total}

angular

^momen-

tum of

23,

^a result that is difficult to obtain

by

^the

usual methods of determinantal states.

14.

Programming

Considerations. -1. STORAGE OF

PARTITIONS. ^- When one sits down to programme any of the

operations

^{of the}

algebra,

^one

immediately

strikes the difficult

problem

^{of how to}

efficiently

remember ^a

partition

or a sum of

partitions.

^The

LE JOURNAL DE PHYSIQUE. ^- T. 30. N° 10. OCTOBRE 1969.

method used is influenced to a

large

^extent

by

^the

machine,

^{and since our} machine is a fixed word

length binary

^{one, an IBM}

360/44,

^{with a} Fortran IV

compi- ler,

^all

partitions

^and coefficients have been stored at

half-word

integers (16 binary bits).

^{Vectors are used}

for

storing partitions

and matrices for ^sums of

parti-

tions. Variable dimensions in the subroutines allow

us to store the coefficient and

partition

^{as a column}

vector of the

length

^{of the}

longest

^vector

likely

^{to be}

required

^{for the}

problem. Any

^{waste storage is}

quite tolerable,

^{since the} storage

required

^{for the}

partitions required

^to evaluate inner

products

^of

Slo

^{is of the}

order of that

required

for the instructions for the Kronecker

product

subroutine.

2. BASIC SUBROUTINES. ^- Ordered

partitions

^{of all}

numbers _up to a

given

^maximum may be

quickly produced

ⁱⁿ

lexigraphical order, by

^a small routine if

we remember which is the last

part greater

^{than one} and the current sum of the parts.

A routine to store a

partition

and coefficient in ^a matrix of other

partitions

^{is useful.}

Ordering

^the

partitions

^{means we} may

quickly

^{find its}

equal,

^{if it}

has _one, and add the coefficients.

Routines to

perform

^outer

products

and divisions of S-functions ^{are a} little more difficult to write since

we must remember the labels for the _squares. When this is

accomplished

^{the rest of the}

programming

^is

more

straight forward,

^{with one rather}

weighty

excep-

tion,

^{how to} get the results out of the computer ^{in a}

tidy

format. The system ^cannot

print

^{a 10 in a}

single

decimal _space and nor are blanks

sightly.

^{With the}

continuous _group characters we know the size of the

partitions

^{as it is}

padded

^{with zeros, but} with S-func- tions the brackets and other

signs

vary in

position.

This

problem

was overcome

by

^{use of an assembler}

output

routine. This routine had several

output

buffers so we could

print

on-line with ^a

typewriter

with _very little loss in machine

efficiency.

15. Conclusions. ^- The results in this _paper have resolved _many of the

ambiguities

associated with the

representation theory

^as

applied

^{to the}

problems

ⁱⁿ

both atomic and nuclear

physics

of the classification of states and

operators

^and the derivation of selection rules.

Computer

programmes have been written to encompass these results and relieve much of the tedium of calculation. Extensive tabulations of results ^are

being published [20].

The addition of

general

methods for the machine calculation of

plethysms

^{and the}

handling

^{of difference}

characters and

spin representations

^{for the}

orthogonal

and rotation _groups should remove many of the remai-

ning problems.

^These

problems

will be discussed in

a

separate

paper.

16.

Acknowledgements.

^{- We are}

grateful to Julian

Brown for assistance with certain aspects ^{of the} pro-

gramming,

^{and to the}

University

^of

Canterbury

^for

the use of their computer.

51

802

REFERENCES

[1]

^RACAH

(G.),

Phys. Rev., 1949, 76, 1352.

[2]

^RACAH

(G.), Group Theory

^and

Spectroscopy, Ergeb.

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^{Berlin, 1965.}

[3]

^CARTAN

(E.),

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[4]

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LITTLEWOOD

(D. E.),

Lond. Math. Soc., 1948, 50, 349.

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^MURNAGHAN (F.

D.),

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Unitary

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LITTLEWOOD

(D. E.), J.

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LITTLEWOOD

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(P. R.)

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(B. G.), J.

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(B. G.), Symmetry Principles

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appendix

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by

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& Sons, ^New York,

1970

(in press).