i (*> i * # i -m —t

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

DEGENERATE REPRESENTATIONSOF THE SYMPLECTIC GROUPS

I. THE COMPACT GROUP Sp(n)

P. PAJASAND

R.

1966PIAZZA OBERDAN

TRIESTE

IC/66/77

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

DEGENERATE REPRESENTATIONS OF THE SYMPLECTIC GROUPS

I. THE COMPACT GROUP Sp(n)t

P. PAJAS*

and

R. RACZKA**

TRIESTE

June 1966

' To be submitted to Proceedings of the Royal Society (London)

* Present address: Nuclear Research Institute, Rez, Czechoslovakia

** On leave of absence from Institute of Nuclear Research, Warsaw, Poland

DEGENERATE REPRESENTATIONS OF THE SYMPLECTIC GROUPS

I. THE COMPACT GROUP Sp(n)

I. INTRODUCTION

Many attempts have been made in the last few years to understand

the properties of physical systems such as elementary particles, the

hydrogen atom, nuclei etc., using the theory of representations of

the underlying symmetry group. The main effort was devoted to the rota-

tion and unitary groups, while the class of symplectic groups did not receive

much attention. This, may be due to the peculiar property of these groups

of conserving an anti-symmetric bilinear form.

Some interest in symplectic groups was raised by remarks of LIPKIN

[1] on possible applications of the group Sp(n#R) to systems of bosons which

do not conserve the number of particles. BUDINI [2] has pointed out that,

using Sp(6, 6) as a higher symmetry group.it is possible to obtain a mass

formula for elementary particles without symmetry breaking. The quest-

ions of symplectic symmetry of hadrons and of the embedding of the harmonic

oscillator in the symplectic group have been discussed in [3] . On the other

hand, in a series of papers [4], the theory of the degenerate representations

of the rotation and unitary (both compact and non-compact)groups, have

been developed. In this work we present the extension of that approach to

the unitary symplectic, groups, i. e., those which conserve both symmetric

and anti-symmetric bilinear forms.

In general, the irreducible unitary representations of a semi-simple

Lie group G are realized as mappings of a Hilbert space yj. (/Cj into itself,

the domain of corresponding functions being some homogeneous space X

of the type

X - 7 G O (1-1)

when G is a closed subgroup of G .

GEL'FAND [5] has proved the important theorem which states that the

number of independent invariant operators in the enveloping algebra acting

- 1 -

in the Hilbert space of functions 'iv ( X) with domain X is equal to the

rank of the space X (and is therefore independent of the rank of the

fundamental group G ), Since we are primarily interested in construction

of representations characterized by the minimum number of invariants, we

can use this theorem to select an appropriate domain A , namely that of

rank one.

In order to select the proper invariant operator we can use the theo-

rem of HELGASON ([6] , Chap. 4, p. 397) according to which the ring of

invariant operators in the algebra (rL of the group Gr > realized on the

space of rank one, is generated by the Laplace-Beltrami operator.

A (X) - -=L= \ 2^ (X) vlfT Pp o- 2»

on X . Here a, M X) is defined by

where q, (X) is the metric tensor on the space X and |£H- I «st [ i f iWn

The operator (1. 2) is actually equal to the second order

Casimir operator Q = %;.• Z ^ ( ??'<£. (y^j of the group

G , provided that the metric tensor Ckj<? ^X) °^ n e space /\ is induced

by the Cartan metric tensor (1. . of the algebra S\ of u , (see [6] , Chap.

X, p. 451).

Then the problem of construction of the most degenerate irreducible

unitary representations is reduced to the problem of determining eigen-

functions and eigenvalues of the Laplace-Beltrami operator on the appro-

priate symmetric space A~ &/&aof rank one.

We select a suitable domain X and solve the eigenproblem of the

Laplace-Beltrami operator on it in Section II. Section III is then devoted

to the study of the properties of the most degenerate representation of the

group Sp(n) obtained in this way. In Section IV we discuss some aspects

of the determination of the series of less degenerate representations of

Sp(n) characterized by two independent numbers. Thus in the present

- 2 -

• : - • ? • ; ! * • • ft *

paper we shall deal only with the case of the compact group Sp(n). We

shall, however, use the results obtained here in forthcoming papers, in

which we would like to solve the following problems:

(i) The construction of a representation space for (most) degenerate

irreducible unitary representations of the non-compact unitary symmetric

group Sp(p, q) determined by a discrete or a continuous invariant.

(ii) The decomposition of the tensor product of two representations of

Sp(p, q) group into irreducible components and the decomposition of the

irreducible unitary representations of Sp(p, q) with respect to compact and/

or non-compact subgroups.

(iii) The investigation and construction of infinite-dimensional represent-

ations of the compact group Sp(n).

n. CONSTRUCTION OF THE REPRESENTATION SPACE

According to GEL'FAND's theorem [5] , the properties of the ir-

reducible unitary representations of a group Cr realized on a Hilbert

space Ti (X) are determined by the geometrical properties of a domain ) \

of functions -V (X) € <3( iX), the domain X being some homogeneous

space.

Symmetric spaces of the type (1.1) with a compact stability group £ro

have been classified by E. Cartan, whereas those with non-compact sta-

bility group have been listed by ROSENFELD [7] . We reproduce in Table I

Cartan1 s list of symmetric spaces [6] for the fundamental group Gr of the

symplectic type. There are also collected the spaces from Rosenfeld's

list, together with their ranks arid dimensions.

- 3 -

Table I

Homogeneous spaces connected with symplectic groups.

Cartan's List

Go - compact.

p n/uu)

Alin)

Rank of A

a

71

~ »

Dimensionof/

71 (W)

Rosenfeld's List

Go - non-compact

X - Gr/&. Rank of X

pn

/

Dimensionof X

X Cti'-O1'*-}

We see that the only suitable candidate for a space of rank one on

which the compact group Sp(n) acts transitively is the space

(2.1)

This space is known to be a quaternionic protective space ([6] ,[7]). But

it is rather difficult to construct a convenient and simple geometrical model

for it. Fortunately, we may use for our purposes the space

(2.2)

which is evidently closely related to the space (2.1). Furthermore, the

space X i s isomorphic to the unitary sphere in the n-dimensional quater-

nionic unitary spaceit

t defined by the equation

%(2.3)

It has been proved by CHEVALLEY [8] and HSIEN-CHUNG [9] that the

group Sp(n) <g>Sp(l) acts transitively on (2. 3) and that its stability group is

Sp(n-l)g>Sp(l).

As is well known, the non-commutative algebra of quaternions GL is

defined as an algebra of dimension 4 over the field g? of real numbers with

a base composed of four elements 1, i, j , k whose multiplication table is

t

I

i

11

i

1 6L' 4

- i {<

J -i

Uk

t

7

-I

(2.4)

Then any quaternion 0 € ( ^ may be expressed either in the form

- 5 -

where X{ (^» V"A)axe real numbers or

I ' ^ + ZzJ (2.5')

where Z{ fl= 1,1) a re complex numbers.

The quaternionic conjugation is the mapping

*t*i£ (2.6)

of Q. into itself. For a detailed treatment of properties of the body of

quaternions as well as for the questions about the relation of symplectic

groups to the vector spaces over the body of quaternions see , for example,

the book of CHEVALLEY[8] .

It is important that the n-dimensional quaternionic unitary sphere

(2. 3) is homeomorphic to the usual sphere in 4n-dimensional Euclidean

space (f\ . Because of it, its properties are rather simple.

Now let us introduce an inner co-ordinate system on the sphere (2. 3).

Let us suppose that we have defined a co-ordinate system on the quater-

nionic unitary sphere (2. 3) of dimension j> < 7t . Let these co-ordinates

be denoted by ^ ( ii - ^ / P ) • Then the co-ordinate system on the

"sphere" of dimension •[> + 1 will be defined by

(2.8)

Now, starting from

= (e^ux^ + e^W^j) (2.9)

we get the co-ordinate system for an arbitrary dimension of the quaternionic

unitary sphere (2. 3) using recursive formulas (2. 7) and (2. 8). This choice

is convenient because there appears in brackets in (2. 8) and (2.9) a general

expression for a quaternion of modulus equal to one. The ranges of varia-

bles Vfc , Vk , $K and 0 k must be chosen so that the co-ordinates (2.7)

cover the space ^ *(•«--1 only once. In this way we introduce on the

space X ^ " 1

- 6 -

n variables € [O,27) C W * - I , ...,

n

n

" •

and (n-1) 2, . . . , 1* )

i . e . , ^n-1 variables altogether.

The metric tensor Qj& (X* ) induced by the metric tensor

of the quaternionic unitary space Q f x ' is given by the symmetric part of

the tensor defined by

f

where

and

ftIn our parametrizatxon the metric tensor <X^» (^ / is diagonal and

therefore the Laplace-Beltrami operator (1. 3) can be represented in the

form:

(2.12)

k(<*-*\~)

where

/ J I• /

and A(X n" '" ) is the Laplace-Beltrami operator on the quaternionic unitary

sphere embedded in the space Q; . For n=l we have

-7-

To find the basis functions for the Hilbert space c* (X) on which the r e -

presentations of the Sp(n) group may be realized, we have to solve the

equation

)

where SL stands for the set of variables {^v ••• > 51* n r . Representr

ing solutions V (SlM) of (2.15) in the form

we obtain the set of ordinary differential equations of second order

(2.19)

(2. 20)

(2. 21)

General solutions of equations (2.19) and (2. 20) a re given in terms of hyper-

geometrical functions as follows

(2.22)

(2.23)

- 8 -

where the eigenvalues wn and h^ are integers and the spectrum of

remaining eigenvalues is given by

(2,24)

and

(2,25)

with positive integers J^ and L .

These solutions are square integrable with respect to the measure

if the following restrictions on eigenvalues are satisfied::

and

L ^ °.V--> L "? I

(

(2.27)

The solutions (2.22) and (2.23) are expressible in terms of the usual d-

functions of the theory of angular momenta [10] . Then the eigenfunctions

of (2,15) are explicitly given by

^

rr<J

where

(2.28)

(2. 29)

and

(2. 30)

- 9 -

The normalization constant NA is then

N * T i n ( V v n 6£k.H)(Ll<+-,2li-'f) . (2.31)

The functions (2, 28) with a given value L = Ln are square integrable with

respect to the left-invariant measure

(2.32)

on the domain X . The explicit expression for the measure djd. is

C

wL L I (2.33)Hence, the set of functions J — (J»i- ) span the Hilbert space

0** i ) c

ji [ A /with the scalar product related to the left invariant measure

(2. 30) by

(2.34)

for any V /C £ OV \A ) . In fact, the space "% (X J i s a

representation space of the group Sp(l) (50 Sp(n) which occurs as a fundament-

al group of the space (2. 2). However, a closer study of the properties of

the Lie algebras of Sp(l) and Sp(n) groups reveals that we can realize ir-

reducible unitary representations of Sp(n) on certain subspaces of the

Hilbert space

HI. THE MOST DEGENERATE REPRESENTATIONS OF Sp(n)

1° Structure of the Lie algebra

The group Sp(l) (g)Sp(n) which acts on the manifold {2. 2) is a direct

product of two simple groups. Therefore its Lie algebra (R decomposes in-

to two commuting subalgebras which we call VS} and i)\^ , respectively.

-10-

The algebra dv^ of Sp(n) Is then formed by the K (a* + 1) generators

which have the symmetry properties

K- H<> <-V< — V--^.: (3-2)The commutation relations of these generators are

( 3 - 4 )

We have collected in Appendix I the explicit expressions for the generators

(3.1) as linear differential operators in quatemionic and complex variables

as well as their connection with the generators of the group Ci.C2.iy.

The algebra (R ± of Sp(l) is generated by the three operators Wg

(X = 1* 2, 3) which have on the manifold A the form

(3-6)

There is a close relation between the operators U.^ and 1 / ^ and UL^k

and XTjck . It is easiest to see from expressions {AI# 22) of Appendix I

which define the generators in te rms of complex variables. The meaning

of the tilde in eqttations (3. 6) then consists in the substitution z* *=z> *•*,

for only K=- - lf •••, - K- , while remaining variables a r e unchanged

( z k - ^ z k and z K - ^ z k for k = X} ••- ? ^ ) .

The commutation relations for the generators (3. 6) are:

j w ; ft-*i«.^ 0.7)

-11-

Throughout this paper we often use, instead of (3.1) and (3. 6), the set of

generators of the complex extension of the real Lie algebras (xK and *Kt

These are especially convenient when dealing with the basis functions (2. 23}

and can be normalized in such a way that they form the Weyl's standard

basis. We define these operators by

( 3 - 9 )

- ' i r t \ (3.10)

and

In Appendix I we give their commutation relations and their explicit form

x in the parametriz-

ation (2.10).

2 Properties of the generators.

As we are using the quaternionic unitary sphere (2. 3) instead of the

quaternionic projective space (2.1) we must be aware of the fact that the

irreducible unitary representations of the group Sp(l) (g)Sp(n) are directly

realized on the space ** ^ h /spanned by functions (2. 28). Nevertheless,

the space "K- (X ) should be reducible with respect to the action of the

group Sp(n). To show this, we use the formulas for the action of generators

of algebras ^ K and KL of Sp(n) and Sp(l) on the basis functions (2. 28),

respectively. They are collected in Appendix II and one can easily see

that the generators have the following properties:

(i) The generators H {p^l, . ^ n ) form the Cartan subalgebra of (k K

and are diagonal in parametrization (2.10). They have the eigenvalues

Mp * r*f. + wtf, (3.12)

-12-

•r ft -s ; . : « r t . ! S > H i ! - . 't & ••'- ii. '.:•-*

(ii) T h e g e n e r a t o r s *~j£?(Prl>'••? a ) c o n s e r v e a l l n u m b e r s !_;. a n d J?'L

a n d a l s o t h e v a l u e of

KA- _— (3.13)lp - mp p,

(iii) All remaining generators conserve the value of

The last property is simply a consequence of the fact that the generator wz

of Sp(l) which has the eigenvalue (3.14) commutes with the algebra K *.

of Sp(n). Therefore, the space A- ( / ) spanned by functions (2.28)

decomposes into subspaces ov ^- C * / o f simultaneous eigenfunctions

of the Laplace-Beltrami operator /X fX *" ) and of the generator *Wj

of Sp(l).

Now the value of M~ is restricted by the conditions (2. 26) and (2.27)

so, that

/ M l ^ L (3-15)

The structure and properties of subspaces <p- ^~ (/ Jstrongly depends

on the value of M . In the case when JM \ - L the subspaces <J[+L(/( j

are irreducible under the action of the group Sp(n) and therefore they can

be considered as representation spaces for a unitary irreducible represent-

ation of Sp(n). Because these representations are characterized by a single

number L , we call them most degenerate representations. They will be

treated in detail in this section. In the case when I Mi ^ L the situation

is not so simple. The space J\. u- {.A J is in this case reducible

with respect to the action of the algebra of Sp(n) and to obtain its irreduc-

ible components one needs further investigation. We have devoted Section

IV to these questions.

3° Unitarity and irreducibility of the most degenerate representations

of Sp(n).

The condition H - + L reduces the two sets of equations

(2.26) and (2.27) to

-13-

- vnf) . tp (p*^...,*) (3.16)

and

V * +#p - / Cp'2y -,*•/3 (3.17)

These are strict conditions on the eigenvalues and they select from the

set of eigenfunctions (2. 28) of the Laplace-Beltrami operator the subset of

functions

(3.18)

Here we have introduced the notation

and

^'i^^^i (3.20)

In the special case / * d we have

We have also put L = L^ .

In the considerations which follow a key role is played jjy the second order

invariant operator !„ which is proportional to the second order Casimir

operator of the group Sp(n). We have found the following connection of

this operator with the Laplace-Beltrami operator A (X "~ j and the^ til

second order invariant operator ~LA of the Sp(l) group which enters the

direct product Sp(l) H Sp(n):

-14-

(3.22)

Here we define

l r = t HZ - 2 [E., f^ + f E ] (3. 23)OC>O

and

ii* (3-24)

while /\ is defined by equation (2.12).

The functions (3.18) are simultaneous eigenf unctions of the operators/\ and jT^ and therefore also of J ^ , the eigenvalues being

and L(L+£n} ^respectively. They span a Hilbert space

*1"' ) d e f l n e d by ^ e scalar product (2.31) with the left invariantmeasure (2.33).

a) Unitarity

The space *K> L ( X / creates a representation space for the

group Sp(n) because for any generator Z ^ £ R^ and any f e %L (A

we have

We shall denote representations of the group Sp(n) related to this space by

DL (Sp(n)) or simply D . They are realized by associating to any

S ( ) t X i n ^ " ' ) uch thatelement Q € Sp(n) an operator X in nL\.^"' ) such that

(3.26)

-15-

for any f { -><L) - ^ ^ " * V>/ ">o ^ ' /W/ ( ^ - J of

JVL^' J« Here i i, is a point of the manifold X and a^JZ is itsleft translation by the element oT1 of Sp(n). - Then unitarity of re-

presentations DL follows immediately from the left invariance of themeasure d,jx(SLj .

b) Irreducibility.

From the explicit form {3.18) of the eigenfunctions I u+ we see

that the structure of the Hilbert space ^> L (. X / i s relatively simple.

Namely, we can decompose >i L (X ) in the direct sum of subspaces as

follows:

where the summation over L^.^and fll goes through

and

(3.28)

(3.29)

/ +respectively. Each of the spaces %/ ^+(X "'') forms a represent-

ation space for an irreducible unitary representation of the Sp(n-l) sub-

group of Sp(n). We see that any representation of the maximal subgroup

Sp(n-1) occurs only once in the decomposition (3. 23). This is illustrated

diagrammatically in Fig. 1.

ftl,

To each point ( L ^ > $K) in the diagram Pig. l.a) there corresponds a

diagram Fig. l.b) which gives the possible values of *Hn and ttiK for a given

* * •

Now, to prove the irreducibility of representations D L of Sp(n) it is

sufficient to prove that, starting from any point in diagrams a) and b) of Fig, 1,

we can by successive application of the generators o? the algebra (\«. of

Sp(n) reach any other point in these diagrams.

It is simple to prove this for Sp(l) (\ - l) because the algebra of Sp(l)

is formed by the three generators H«g» and H ; the former act as step

operators while the latter is diagonal. The presence of the "stopping"

factor / ( X ? m T i T in the formula (All. 3) of Appendix II for the action

of the generators E -o'e* assures us that starting from some point of Fig. 1,

diagram b) we can reach any other point on the diagranxand only these points.

This means that the irreducibility of representation Du ^ 1_ = J?1) of Sp(l)

is proved. Similarly one proves the irreducibility of the most degenerate

representations for the Sp(2) and Sp(3) groups. Now let us assume that we

have proved the irreducibility of D L for the group Sp(n-l). It means that

the spaces d< i ^+ ( * ) are irreducible with respect to the action of the

generators of the algebra MV^^L of Sp(n-l).

Now, to prove the irreducibility of /V L (_ A ) space with respect

to the actions of the algebra ^ ^ of Sp(n) it is sufficient to consider only

the action of generators E + ae? > ^^v-^n- i a n d ^ ^e1 + ^v-i '

(See also Appendix II). The corresponding formulas are formulas (All, 11-

14) and (All. 3) of Appendix II. We see that the considered generators con-

serve the conditions (3.16) and (3.17), Moreover, the stopping factors are

always combined in such a way that relations

4 > O ; Ln.L >oand

/ rnn + m» U 4 ; I *.., + v^n., I ^ L ^

are fulfilled.

This procedure enables us to prove the irredueiMlity of representations

X> L of Sp(n) by full induction.

-17-

4° Properties of the most degenerate representations of Sp(n).

/up L / w l/n-f\The basis functions (3.18) of the Hilbert space A\.,, (A Jare

characterized by n numbers L* and n numbers Hjf O'-V-/*?)

The former are related to the eigenvalues of the set of the second order4 (i)

invariant operators -ly> of the chain of subgroups

p p p (3.30)

These eigenvalues are

Numbers M/" are then eigenvalues of the generators -H& of the Cartan

subgroup of Sp(n).

Therefore, the set of commuting operators is explicitly formed by

hl~2<n (2.32)

operators:

The number (3. 32) is reasonably small compared to the corresponding

number in the case of a non-degenerate series of representations of Sp(n),

where it is

This may be. of particular interest from the point of view of physical

applications, because usually we would like to have the smallest possible

number of invariants for characterization of a given physical state. The

fact that all Cartan subgroup generators are diagonal,which is due to the

parametrization we have employed, is also convenient for it makes it

possible to relate each of these generators to an additive conservation law.

Another property which is a direct consequence of the choice of the

parametrization of the domain X is the pattern of the decomposition

-18-

of the Hilbert space dV L ( . / ) into the aubspaces o\-/_ ^ (A

on which the subgroup Sp(n-l) acts irreducibly. One can easily find a

parametrization in which the representation 2)L of Sp(n) decomposes in

a similar way with respect to a subgroup Sp(n-k) with arbitrary k (*" ^ £

In Appendix III we give the detailed calculation of the highest weight

of the representation of Sp(n). From it follows, that the representation

T>u corresponds to the representation D(L, (?,...,£?) in the notation used

in 111] , or it can be represented by a one-row Young tableaux (see [12] )

and therefore may be interpreted as a fully symmetrical representation

of Sp(n).

More detailed examples for the lowest representations of Sp(2) and

Sp(3) are given in Tables I and II of Appendix IV.

IV. THE LESS DEGENERATE REPRESENTATIONS OF Sp(n) AND THE

REPRESENTATIONS OF THE GROUP Sp(l) H Sp(n),

1° The less degenerate representations of Sp(nX

The condition

| M " | - wwsi < L

imposes a restriction on the eigenvalues, which can be conventionally

written as

But from (2.27) it follows that

|H"| 4 L -il ( 4 - 2 )

where now k may be one of the run

We can again represent the possible values of L^y and ** by points in

the net of Fig. 2

- 1 9 -

Fig. 2

From the formula (AII#13) of Appendix II we see that the operator J-Y and

therefore also the second order invariant operator J» of Spfn) are not

diagonal on the space <A M_ ( / ) . Generally the operators xi and

Tn when acting on functions f2. 28) conserve, besides all the numbers L^

and ^A ( ^ V ' i i ) > also the numbers M,,4" (/> =• ^ •• ,n) and M

but generally they change the values of MA 0>= I_, , -., ^ ) . The last

numbers are eigenvalues of operators f> which do not belong to the set

of commuting operators of the algebra (jln of Spfn). Therefore, it is

necessary to diagonalize the operators -i- and -Ln on the subs paces

3^ ufu ( X "") ' This leads to the usual procedure of diagonalizationHA i"~

of matrices by which the generators are represented on a given space.

In order to see the structure of the space <jl ^-{A /let us

first investigate the points for which Ln_t •= O . Then we have all other

numbers equal to zero except / h and M^ . The value of M^ is fixed

by

<4 -4 )

There fore the condition (4. 4) defines the function Yul i"''°J *'°'"'

The eigenvalue of X will then be

(4.5)

-20-

This means that functions with Lr)_i » 0 and different values of -U in

- (Y *"') belong to different irreducible representations of Sp(n).

We denote the eigenvalue of X, by

we get

t = I M I ; I H - I

and then from (4. 5) and (4.1)

., L (4.6)

Now, taking into account the generators of the algebra 31^ we see that

to the same representation as the point ( L , O ) in diagram

must also belong the points ( L i 4-} "I j . Proceeding in this way we get

the following decomposition of the space ^f [Y n / into the sub-

spaces which are irreducible under the action of Sp(n):

L

This is illustrated diagrammatically in Fig. 3

M0L

Fig. 3

Of course, the set of functions which span the space t

by the diagonalization of the operator Jf in all subspaces

Therefore, the function £f M// fi> * ^

operators:

obtained

kkyis the simultaneous eigenfunction

of the set of

and is generally expressed as a certain linear combination of functions

-21-

pwhich span the space <?v H * n-

On the spaces <Kr ^-(X "') one can introduce the irreducible

unitary representations of the Sp(n). Unitarity of these representations

can be proved in the same way as it was in the case of the most degenerate

representations.

The highest weight of a representation 3>£- realized on the space

~ K (X J * s c a l c u l a ted *n Appendix III. Then the representation

£. can be denoted according to the notation of [11] by

,<>(4.9)

which corresponds to the Young tableaux with two rows defined by the

symmetry scheme

-— {~*vl

JThe representations realized on spaces and

are equivalent because they have the same highest weight.

The diagram Fig. 4

T

(4.10)

where

Fig. 4

- 2 2 -

corresponds to the pattern of decomposition of the representation T)^

of Sp(n) in respect to the irreducible representations of the maximal sub-

group Sp(l) H Sp(n).

In Appendix IV we give detailed examples of lowest representationsf '

of this series (which we call less degenerate) of the groups Sp(2) and Sp(3).

2° Representations of Sp(l)(x)Sp(n).

Finally, we should remark that the full set of functions (2. 28) span a

Hilbert space % (X^^ ) which acts as a carrier space for irreducible

unitary representations of the group Sp(l)<3£>Sp(n).

on L / y l ( .H -3 \

The corresponding decomposition of the Hilbert space av \ A /

into subspaces is then given by

/ / — ( "* * W•'•/l^i ^l( Wj, VJ^ * • / #

(4.11)

where the summation is restricted by conditions (2.26) and (2.27). There

is no need to repeat the same kind of consideration as above to prove the

irreducibility and unitarity of the representations DL of s'p(l)<g>Sp(n) on

this space. Since Sp(l) GDSp(n) is a direct product and we have treated

Sp{n) in the previous sections, it is only necessary to use generators from

both (j V and £K algebras of Sp(l) and Sp(n), respectively, to prove the

irreducibility of representation D . Representations DU of Sp(l)®

Sp(n) decompose into the representations of the subgroups Sp(l) and Sp(n)

according to the formula

D . ,(4.12)

as may be easily verified.

- 2 3 -

V. CONCLUSION

The main results can be summarized as follows:

(i) A set of harmonic functions which span the representation space

for the most degenerate unitary irreducible representations of the compact

symplectic group Sp(n) has been found. These representations are charcter-

ized by a single number which is related to the eigenvalue of the second

order Casimir operator of Sp(n). They can be also described by a one-

row Young tableaux.

(ii) Besides this, the series of unitary irreducible representations of

Sp(n) characterized by two independent numbers L and L has been

obtained. These numbers are related to the eigenvalues of the Laplace-

Beltrami operator A (X " 7 on the guaternionic unitary sphere X

and of the second order Casimir operator of Sp(n), respectively. To these

representations corresponds the Young tableaux with two rows.

(iii) The number of operators in the maximal set of commuting operators,

the eigenvalues of which characterize the basis functions of the represent-

ation space, is in the case of the most degenerate representations

A/,-in (5.1)

and in the case of the less degenerate representations it is

^ , \ n ^ (5-2)

These numbers are sufficiently small when compared to the corresponding

number in the case of non-degenerate representations of Sp(n);

This underlines the importance of the degenerate representations for

physical applications.

(iv) In the parametrization introduced On the quaternionic unitary sphere

A , the generators n^ which form the Cartan subalgebra are all

diagonal which is again useful for applications.

-24-

,.i. i,. ".,.: • ,

(v) Finally, the patterns of the decomposition of the given irreducible

representation of Sp(n) with respect to the maximal compact subgroup Sp(l)

63 Sp(n-l) of it,occur due to the parametrization.

ACKNOWLEDGEMENTS

The authors would like to thank the IAEA and Professors Abdus

Salam and P. Budini for the hospitality extended to them at the International

Centre for Theoretical Physics in Trieste. It is a pleasure to thank

Dr. J. Fischer for many helpful suggestions and Drs. J. Niederle and

A. Bartl for interesting discussions as well as Dr. R. Anderson for kindly

reading the manuscript.

-25 -

APPENDIX I

Algebra of Spfn) and of Spfl)

In order to obtain the algebras of Spfn) and Sp(l) groups, let us

consider an infinitesimal symplectic transformation in the unitary n-dimen-

sional quatemionic space Q. *

A"

The necessary and sufficient condition for the n x n matrix of quatemionic

elements A to be symplectic is

A A4 - A4A - X (AI.2)t r

(i'

where

We point out that the dagger here represents the quatemionic conjugate defined

in Section JII, and the transposition of matrix A . Representing any element

of A in the form

and considering the real parameters a s t , & s t 1 cs(- and d. . ^.S;t = j v . . .

as infinitesimally small quantities, we obtain,imposing condition (AI. 2)7

(AI. 5)

(AI.6)

or _

A = ^ ^Let us now define four formal.linearly independent quaterBionic quantities

- 26 -

W l ^ H ' i - H • - ! • • • * ^

' 7 )

, •<

and define the representation of the group Sp(n) as a transformation in the

space of functions of these variables determined by

for any A € Sp(n). Defining the formal derivatives by

J (AI. 9 )

we can expand the right-hand side of equation (AI.8) into a Taylor series.

Then the generators of the Sp(n) group are obtained if we consider the one-

parameter subgroups of Sp(n). This procedure leads to the following set

of generators:

- 27 -

(AI.10)

Here we have introduced the notation that for example

represents the expression

t +

The commutation relations (3.3) - (3.5) of the generators w f and U^,

which are given in Section III, are easily verified using the definition (AI.9)

of quaternionic derivatives, or expressing generators UJ^ and T££ in

variables <£, # £,. #* and jt* as follows:

j

* ^ 7

- 2 8 -

Now we may express these generators through those of U(2n) given in [4]

as

* *A (AI.12)

We see that

Because the algebra %, of the Sp(n) group is the compact real form of the

algebra of (the complex group un ,we can easily find the Weyl's standard

basis K* , E tae > £*lp +F » -i$'$%°t ^*\ by using the generators (AI.10),

This relation is given by formulas (3.8) - (3.11) of Section HI.

The commutation relations of these generators are

J - /v t f i e , ^ (Ai.i5)

where oi> are roots and N^A = ^ .^ ^ 0 only if ot + £ is also a root.

We obtain a Dynkin diagram

-29-

of the group Ch tfwe put

• (AI.16)

where &i > **" ? **- are unit Vectors which form an orthonormal basis in

the root space.

Generators £ + * play the role of raising and lowering operators

when acting on the basis functions (2. 28) of the space #i (X :) while the

generators H^ form a Cartan subalgebra. In the following, we present

explicit expressions for these generators as linear differential operators

in the parametrization of the manifold X ^ *~X we have introduced in

Section II. Similarly, the three generators of CL which correspond

to the Sp(l) group are defined by

W ' " (AI.17)

where »

* (AI.18)

In parametrizationization of X "*v"i we now have

and for t < Jt we have

-30-

(AI.21)

and

E * 2 *

t 9

- 3 1 -

Here

A(AI.23)

and

Oat OJL fr>\*I-I

-X<l

— c<s<>(AI.24)

The algebra of Sp'(l) in Sp'fl) <g>Sp(n) is then formed by three generators

(AI.25)

and

(AI.26)

It is rather easy to show that these generators commute with the algebra of

Sp(n).

- 3 2 -

APPENDIX II

Actions of generators on basic functions.

To prove irreducibility of the representations D (X ) and

' we need explicit formulas for the action of different generators

of the algebras Rj_ and <RK of Sp(l) and Sp{n), respectively. However,

we do not need to know this action for all generators of Sp(n) because if we

know, say, the action of E ^ + g» , Ej, 7g» and Eyg1 for any

b= a} ..., *\ , we can use the commutation relations (AH5) to get, for example,

\g + £• for any k< t> • For this reason we give in this Appendix the

formulas for the action of the generators E+<f +•£ , Et » _•£» and E+^

on the basis functions (2. 26).

As may easily be seen from the explicit expressions of the genera-

tors in Appendix I. the generators ,Et^ do not change either values

Lt ((c = 3,...,ti) or ^ K (K= ij • -•."), while £.5* . ? and £4^ _£» change

in these sets of eigenvalues only L,,,^, x^ and JJ*,_j_ . All the generators,

except those of H ^ which form the Cartan subalgebra, change the values of

*v\ . Tvt i.. Av\. . andi«. , . Therefore, we label the eigenfunctions in what

follows only by the eigenvalues Lp=L , Z^.j * i! $.a - -t , fj>-i £ £ ,

t __ , —_— -t—/ . Thus we have:

Hp

LJ VL'L')fi£i' r -^ v ^ ' / V ' ( A I L 2 )LJ Vnp 1 / |_

L,L-A tl\ 4 t—• r v L>L>)£'£;

^ {An.4}

-33-

{AH. 5)

7

( t ± ,* y,LliteL

-34 -

Here (AH. 6)

(All . 7)

(AH. 8)

^C

The coefficients df and d* are the same as C* and Cj , respectively,

if one substitutes Jt*, J and K' for i , m and * , respectively.

In the case of the representations 3> ( X ^ ' *) of Sp(n), when

L'.L'+t!

and

we have

This reduces expressions (AH. 5) and (AIL 6) to the following form:

4

(An.

(AH. 12)

-35 -

The action of the invariant operator I , connected with the Sp(l) component

of Sp(l) Ef Sp(n) is given by the formula

z: y

where

(All. 13)

c r { A I L

-36-

APPENDIX in

Calculation of the highest weight.

In a representation X> of a semi-simple Lie group <3r of rank n.,

in which the representives I>(H;Jof the Cartan subgroup generators \ are

diagonal, the eigenstates and eigevalues of lifW;) are defined by

D0O4 «>*,•*,

The set of eigenvalues lv*\\ may be considered as a n-dimensional vector

in the so-called weight space [13] . A weight y£ is called higher than

another weight 'to ' if the first non-vanishing component 'ht - yv\ ' is a

positive number. The weight A which is higher than any other weight in

a given representation is called the highest weight. As is well known [14],

the highest weight fully characterizes the irreducible representation of a

compact group. We shaU denote its components by A .

It has been proved by Cartan that there exist v\. fundamental weights

A , . . . ; A and that any highest weight A is a linear combination

1 Acu) . 2)

where )• ^ are non-negative integers. Moreover, these fundamental

weights are uniquely defined by the root system of the algebra of

G; from the system ; io</ , . . . ,o( ] of simple roots,we can calculate

the Cartan matrix / A j : i by

A,. - ^ ^ ' * (Am.3)

and then the fundamental weights a r e given by

Uw (Am.4)

where A* i s the inve r se m a t r i x to j /^j ;] . Usually, an i r reduc ib le r e p r e s e n t -

ation of the group Q- i s denoted by ^(X^t •• >^^) . This notation

appea r s in column 2 of tab les II and III and throughout Sections IQ and IV.

-37-

Let us calculate the coefficients A L , ... , 1 ^ for the representations

of the group Sp(n). The eigenvalue of the generator

is clearly

* ^v.

/**tt

To this value corresponds

~ T

Hence, there is a highest possible value of X^.t given by

L- T

(Ain. 5)

(Ain. 6)

So the highest weight will be that with the highest value of J?K . From

Fig, 4 we see that the highest possible value of $^ is

(Am. 7)

(Am. 8)

(Am. 9)

But when (ADI. 7) and (Ain. 9) hold simultaneously, it is easy to see that

"2

So we have

L..., / \ ^ ) « (of ..v

Now in the case of group Sp(n) the Cartan matrix is given by

•2 ~4 - - • O O O

0

O

0oo O -2 X

(Ain.

(Ain.

(Am. 12)

-38-

if we use the definition of o(Jl) ( { a l,..., h) given in Appendix I (A1.16).

Then the set of fundamental weights of Sp(n) is

" * *" " ' • * (AIII.13)7\ C M " l

- ) •f" x °*- • &i -f •

This gives in the case of the representation Dh of Sp(n)

(AIH.14)

Therefore, we have obtained the resulting formula

(AIII.15)

as indicated in Sections IH and TV.

-39-

APPENDIX IV

Lowest representations of Sp(2) and Sp(3)

In this Appendix we consider the lowest representations of Sp(2)

and Sp(3) which have been obtained in this paper.

In Tables I and II we give

1° The comparison of our notation 3> of the representation of Sp(n)

and Sp(3); respectively with the notation based on the highest weight com-

ponents and the Young tableaux, as defined in [ 11] and [12] .

2° The decomposition in respect to the maximal compact subgroup.

This is given by diagrams in the last columns of both tables where at each

point in brackets stand the dimensionalities of the corresponding represent?

ations of the subgroup Sp(l) S Sp(2) and Sp(l) <g>Sp(l), respectively.

In Tables III and IV we give the list of the basis functions for the simplest

less degenerate representations of Sp{2) and Sp(3), respectively. For the

Sp(2) group we use the denotation Y \ x[ i for the original functions

of the set (2. 28) and ^- ] »,+ ^+ for the new orthonormal basis functions.

( ( eIn the case of Sp(3) this denotation is / H f f _ and

' 2 Y respectively.

We have selected from the class cf equivalent representations those

which have H - + L . In the last column of these tables is the number

of functions of the corresponding type, which differ by the eigenvalues M

(s = 1, 2 resp. 3).

-40-

Table II

Lowest representations of Sp(2)

D t

Z.D o

<

<

<

JL

D(XX.A2)

(1,0)

(2,0)

(0,1)

(3,0)

(1,1)

(4,0)

(2,1)

(0,2)

[10]

[20]

[11]

[30]

[21]

[40]

[31]

[22]

Dimension

4

10

5

^0

11

»

35*

14

Young tableaux

J

Diagram (^ 4 )

1!

(

>v fa i i

X / , '" fl

Aa

- 4 1 -

Table in

Lowest representations of Sp(3)

Dimension Young tableaux Diagram (*-j

(1,0,0) 1100] 6

(2,0,0) [200] 21ifi.J*)

D (0,1,0) [110] 1±

(3,0,0) [300] 56N^rSl

(1,1,0) [210]

n.<£,£)

(4,0,0) [400] 126

(2,1,0) [310] 189

(0, 2, 0) [220] 22J*.

Ci.D

-42-

Table IV

Basis functions for the representation spaces of the lowest less degenerate

representations of Sp(2)

Represent

L

D 3

1

List of basis functions

0 l / l j 0,0 yi- 0,0

0 ' O f o ~ I 0, Oj 0,0

f yi;°ii v i ' °« 1

1 y V'1 t y i i0'4

i Y^; - Y^O

No. of

tions

1

4

2

2

6

6

3

4

3

8

9

8

- 4 3 -

Table IV (Continued)

List of basis functionsNo. offunc-

tions

o yh,o,oyh,' o , 0

O,O; o,O

0

0(YH ; ; , t , , - Y,t;

Table V

Basis functions for the representation spaces of the lowest less degenerate

representations of Sp(3)

Represent.

0

0 1 C,

0 ' fij

' \7^0 ' 0,

o 1

o,a

List of basis functions

\ c,o,oto,ofo

f r vi ' V ' HjT, 3, h/j -1 o, i 1 Hi^o, Hf. A, o, -i j

A I fyz,'lioi/i'i y1^)0,^" )

No. of

func-tions

i-i

4

4

1

-44-

Table V (Continued)

List of basis functions No. of

functions

,1,0,0

1 O,o,Hi* ' 0,0,^0,0,4

^ . ^ 0,-1,0

- f . 1 V;2,<U

D

>,0

8

-f,

W^v, - Wi 0,0,4

M

-45-

FOOTNOTES

1) Note that sometimes in the set of invariant operators there appear

operators which are not elements of the enveloping algebra, (For

an explicit example see [4] ).

2) The rank of a space X - &{ % i s defined as the number of in-

variants of any two points x,o- £ X with respect to the action of

the fundamental group G" on A .

3) Here [a] means the integral part of a defined as usual.

4) We are limiting ourselves only to the case H~=+L because the

choice M *= — L leads only to the change of sign in the r . h. s.

of equation (3.16) which means changing the sign at any i/^ in lower

indices of functions (3.18). The corresponding representations

a re equivalent.

-46-

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[11 H. J. LIPKIN, Lie group for pedestrians (Amsterdam, 1965).

[2] P. BUDINI, private communication,

[3) H. BACRY, J. NUYTS and L. VAN HOVE, Nuovo Cimento 35,

510 (1965)j

R. HWA and J. NUYTS, Group embedding for the harmonic

oscillator. Institute for Advanced Study, Princeton, NJ, (1965)

[4J ' N. LIMIC, J. NIEDERLE and R. RACZKA, Discrete degenerate

representations of non-compact rotation groups, ICTP, Trieste,

preprint lC/66/2.

J. FISCHER and R, RACZKA, Discrete degenerate representations

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[6] S. HELGASON, Differential geometry and symmetric spaces.

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[8] C. CHEVALLEY, Theory of Lie groups - I , Princeton University

Press (1964).

[9] WANG HSIEN-CHUNG, Ann. Math. 55, 177 (1952).

[10J M. E. ROSE, Elementary theory of angular momentum, John Wiley

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[11] M. KONUMA, K. SHIMA and M. WADA, Suppl. to Progr. Theoret.

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-47-

[131 R.E. BEHRENDS, J. DREITLEWt C. FRONSDAL and B.W.LEE,

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-48-

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