Ulam Quarterly Volume Numb er

Duality in Repre s entation Theory

W H Klink

Department of Phys ics and Astronomy

Department of Mathematics

The Univers ity of Iowa

Iowa City Iowa

Tuong TonThat

Department of Mathematics

The Univers ity of Iowa

Iowa City Iowa

Ab stract

The SchurWeyl Duality Theorem motivate s the general denition of

dual repre s entations Such repre s entations ar i s e in a very large class of

groups including all compact nilp otent and s emidirect pro duct groups as

well as the complementary groups in the phys ics literature and the re ductive

dual pairs in the mathematics literature Several applications of the notion of

duality are given including the determination of p olynomials and op erator

invar iants and the di scovery of unusual dual algebras

Intro duction

In I Schur prove d a theorem Sc that was later expande d by Weyl

in a form We that i s now calle d the SchurWeyl Duality Theorem Thi s

theorem which i s br iey reviewe d in the next s ection relate s the repre s en

tation theory of two groups the general linear group and the symmetr ic or

p ermutation group It i s the rst theorem as f ar as we know that clas

s ie s up to i somorphi sm all irre ducible repre s entations of one group in

terms of all irre ducible rational repre s entations of the other As we shall

show such a phenomenon in which repre s entations of pairs of groups are

relate d tur ns out to b e quite general It include s the complementary pairs

of groups intro duce d in the phys ics literature by Moshinsky and Que sne

MQ and the re ductive dual pairs intro duce d in the mathematics literature

by many mathematicians Ge GK KV Sa and e sp ecially Howe Ho

All compact groups can b e wr itten as dual repre s entations and it s eems

Duality in Repre s entation Theory

likely that all nilp otent and s emidirect pro duct groups can b e viewe d as

dual repre s entations although thi s remains to b e proven

There are many applications of the theory of dual repre s entations

First knowing that the repre s entations of two groups are dual to one an

other allows one to get the algebras generate d by the group actions f rom

which information can b e obtaine d ab out the commutants One rather sur

pr i s ing connection of such dual algebras i s with the theory of p olynomial

invar iants

0

Second given two groups G and G whos e repre s entations are dual the

re str iction of the repre s entation of G to a subgroup H may give r i s e to an

0 0 0

extens ion of the repre s entation of G to a group H containing G Such a

0

s eesaw phenomenon thi s term was intro duce d by Kudla where in H i s

dual to H give s cons iderable ins ight on the nature of the re str iction of G to

0 0

H or H to G As will b e shown it i s p oss ible to view the decomp os ition

of tensor pro ducts in thi s context and us e the dual pair structure to re solve

the multiplicity problem

Finally giving a repre s entation of a group along with its sp ectral de

comp os ition can lead to unusual repre s entations of a dual group or even to

a dual algebra whos e structure might not b e known or conveniently charac

ter ize d In the next s ection we will give a general denition of dual repre

s entations and provide a numb er of example s to illustrate the generality of

the denition The third s ection will then give a numb er of applications of

the notion of duality

x Denitions and Example s of Dual Repre s entations

To motivate the denition of dual repre s entations we will formulate

the SchurWeyl Duality Theorem in a somewhat unf amiliar way so that

z

nN

n

C denote the space of all it can eas ily b e generalize d Let P

nN

p olynomial functions F on C which sati sfy the covar iant condition

d

nN

F dZ det dF Z d Z C

d

nn

Let S denote the symmetr ic group of degree n and GLN C denote the

n

complex general linear group of order N A joint action of S GLN C

n

on P i s dene d by

L Rg F Z F Z g

nN

F P S g GLN C and Z C

n

Then the SchurWeyl Duality Theorem can b e state d as follows

Duality in Repre s entation Theory

nN nN t

Let Z C and dZ exp tr ZZ dZ and s et

nN

H F C

Z

nN

ff C C f entire and jf Z j dZ g

nN

C

0

Let G GLN C and G GLn C and dene

0 0

Lg f Z f g Z

Rg f Z f Z g

P

nN

Then F C i s irre ducible I and the repre s entation L Rj

I

0

where denote s b oth a repre s entation of G and G in the class of irre ducible

0

repre s entation of G and G lab ele d by an r tuple of integers

r

sati sfying r minn N

r

Generalization of Example No Let H b e the same as in Ex Let

Let p p b e integers such that G GLN C GLN C

m

z

m copie s

p p p n and let

m

0

G GLp C GLp C

m

z

m

nN

Partition Z C in blo ck form as

Z

p N

i

with Z C i m Z

i

Z

m

nN

then G acts on C via

Z g

Z g g g GLN C

m i

Z g

m m

0 nN

and G acts on C via

k Z

k k Z k GLp

m i i

k Z

m

m

0

The s e actions induce a repre s entation of G G on H dene d by

L Rk k g g f Z

m m

f k k Z g g f H

m m

W H Klink and Tuong TonThat

0

Theorem The representation L R of G G on H is decomposed

0

into irreducible subrepresentations labeled by the double signature

G G

0

0

is an irreducible representation of G indexed by an ntuple of where

G

integers of the form

m

m

M M M M M

p p

1 m

i i i

with M M i m and is a tensor M

p

G

i

product of irreducible representations of G indexed by the same ntuple

of integers Moreover each irreducible subspace of this orthogonal direct

0

sum decomposition of H is an isotypic component of both and

G G

See KT and KT

Example i s a typical example of dual or complementary pairs

as dene d by M Moshinksy MQ and generalize d by R Howe Ho An

example of the s eesaw phenomenon ar i s e s if the subgroup H of G i s

0

chos en to b e SO N the dual to H i s H S pm R acting on H as

in Example with n m Note that the irre ducible repre s entations of

S pm R are innitedimens ional metaplectic repre s entations See KV

GK and MQ

x Applications of Dual Repre s entations

0

The invariant theory of block diagonal subgroups of GLn C Let G

GLp C GLp C and H b e the same as in Example of Section

m

nn

Let n p p and cons ider the algebra S C of all p olynomial

m

nn

functions on C Then the adjoint repre s entation of GLn C give s r i s e

nn

to the coadjoint repre s entation of GLn C on S C The p olynomials of

nn

S C that are xe d by the re str iction of thi s coadjoint repre s entation to

0 nn

the blo ck diagonal subgroup G of GLn C form a subalgebra of S C

0

calle d the subalgebra of G invar iants Us ing the work of Pro ce s i Pr on the

0

dual G of G and a Frob enius recipro city theorem relating the dual action

0

of G and G on H give s the following

nn

Theorem See KT If a matrix X C is partitioned in block

form as

X X

m

X

X X

m mm

where each X is a p p matrix i j m then the subalgebra of

ij i j

0

al l G invariant polynomials is nitely generated by the constants and the

functions of the form

X X TraceX

i i i i i i

q 1 2 3 1 2

Duality in Repre s entation Theory

for al l i m k q

k

The resolution of the multiplicity in the decomposition of tensor products

of group representations Example and Theorem are typical of the

way the s eesaw phenomenon i s us e d In thi s example if we cons ider

the re str iction of the repre s entation R of G GLN C GLN C

z

m

to the diagonal subgroup H fg g g GLN C g then we have

m

the problem of decomp os ing an mfold tensor pro duct M M

i

of irre ducible repre s entations of GLN C where each M i m

i s the s ignature of an irre ducible repre s entation of GLN C In general

the sp ectral decomp os ition of thi s tensor pro duct involve s multiplicity To

re solve thi s multiplicity H invar iant dierential op erators or generalize d

Cas imir op erators are intro duce d and the ir e igenvalue s are us e d to lab el

dierent copie s of the same irre ducible repre s entation of H o ccurr ing in thi s

decomp os ition The s e H invar iant dierential op erators are relate d to the

p olynomial invar iants in Application s ee KT and KT

Thi s leads to the following general s etup Let G b e any re ductive Lie

nN

group with the holomorphic repre s entation on the Fo ck space F C of

Example dene d by Rg f Z f Z g and cons ider a holomorphic

nN

irre ducible repre s entation of G on a subspace V of F C If there i s

G

0 0

such that the i sotypic comp onent a dual action of G on a subspace V

0

G

0 0

in general the action of G i s much larger than just the left V i s V

0

G G

0

action unle ss G GLn C and often one only knows the Lie algebra ac

0 0

tion of G then the action of G may b e us e d to re solve the problem of the

re str iction of the repre s entation of G to a subgroup in the following s ens e

Cons ider a subgroup H of G with an irre ducible repre s entation on W

H

0 0

Since H i s containe d in G its dual H contains G and has a repre s enta

0

such that the i sotypic comp onent of the dual repre s entation in tion of W

H

nN 0

F C i s W We then have the generalize d Frob enius recipro city W

0

H H

theorem

0 0

The multiplicity dim HomV W of the irreducible representation

0 0

G G

0 0 0

V in the restriction of W to G is equal to the multiplicity dim Hom

0 0

G H

of the irreducible representation of H in the restriction of W V V

G H H

then generalized Casimir operators dene d in V to H If dimV

H H

the same way as ab ove can b e us e d to break thi s multiplicity

Construction of bases for irreducible representation spaces Dual repre

s entation theory can also b e us e d to nd bas e s for s imple Gmo dule s We

starte d with the s imple example of Gelf andZetlin bas e s for GLn C by

cons ider ing the problem of decomp os ing tensor pro ducts of repre s entations

of GLN C in Example of Section In Theorem if the tensor pro d

ucts of irre ducible repre s entations of G GLN C GLN C

z

m

i

i

for all i cons i st only of s ignature s of the form M M

z N

W H Klink and Tuong TonThat

m then us ing a formula of Weyl the sp ectral decomp os ition of the

i

tensor pro duct of thi s particular M with a general s ignature i s multiplicity

f ree Thus if we cons ider the coupling scheme

m

M M M M

0

then in the dual repre s entation G GLn C we have a corre sp onding

chain of subgroup repre s entations GL C GL C GLn C

which i s preci s ely the Gelf andZetlin chain In KT thi s construction i s

carr ie d out completely to nd Gelf andZetlin bas e s for irre ducible repre

s entation space s of GLn C If dierent coupling scheme s are us e d then

one can construct dierent typ e s of bas e s A s imilar construction for the

Gelf andZetlin bas e s for the chain SO SO SO N can b e

carr ie d out as follows It i s known that succe ss ive re str ictions of an irre

ducible repre s entation of the chain of subgroups SO N SO N

SO i s decomp os e d into multiplicityfree subrepre s entations but

the diculty i s that o dd and even orthogonal groups have a very dierent

structure So it i s natural to cons ider the chain SO N SO N

M

SO N s et G SO N H SO N and let V b e

G

m

an irre ducible Gmo dule and W b e an irre ducible H mo dule It i s easy

H

m M

to compute the multiplicity of W in the re str iction V of G to H

H H

0

Now the dual G mo dule i s the metaplectic repre s entation of S pn R so

0

we nee d to calculate the dual H mo dule whos e rai s ing op erators will s end

m

M

W into the inters ection of the i sotypic comp onent I with the i sotypic

H

m

comp onent of I Again there i s multiplicity which can b e re solve d with

generalize d SO N Cas imir op erators A s imilar construction also can

b e carr ie d out for the compact symplectic groups further the metho ds us e d

to decomp os e tensor pro ducts of the GLN C groups can also b e applie d

to the orthogonal and symplectic groups

Construction of new representations and algebras via duality Us ing the

notion of dual repre s entation can lead to repre s entations of var ious groups

or algebras which are generally dicult to class ify The pro ce dure can b e

de scr ib e d as follows Let K b e a compact group and let b e a nite

dimens ional repre s entation of degree N of K on a complex vector space V

The manyparticle symmetr ic Fo ck space LV i s dene d as

1

X

sym V V LV

z

n

n

where V V sym i s the nfold symmetr ize d tensor pro duct of V If

z

n

j i s an e fe e g i s an orthonormal bas i s for V then e

sym i N i

n 1

orthogonal bas i s of the nparticle subspace of LV The corre sp ondence

Duality in Repre s entation Theory

N

b etween LV and the space F C given in Example of Section i s

given by

Z j Z e e

i sym i i i

n 1 n 1

N

so that an action of K on F C i s given by Rk f Z f ZD k f

N

F C where D k i s the matr ix of k relative to the orthogonal bas i s

N K

of op erators on F C which commute fe e g The algebra A

N

V

with D k k K i s in general an innitedimens ional algebra which has a

CartanWeyl structure with diagonal rai s ing and lower ing op erators Thi s

algebra i s dual to the algebra generate d by the op erators D k k K and

its repre s entations are indexe d by the repre s entations of K app ear ing in the

symmetr ic tensor pro ducts The explicit construction of s everal algebras

K

i s given in K and K A

V

x Conclus ion

In thi s article we have given a numb er of example s and applications

of the notion of dual repre s entations We have however only scratche d

the surf ace of what we think i s a very r ich structure encompass ing a very

large class of group repre s entations and relate d sub jects Thi s make s us

appreciate even more the s imple idea initiate d by Schur and Weyl

Reference s

Ge S Gelbart Examples of dual reductive pairs Automorphic Forms

Repre s entations and Lfunctions Part I A Borel and W Cass el

mar e ds Pro c Symp Pure Math Vol Amer Math So c

Providence RI pp

GK K Gross and R Kunze Bessel functions and representation theory

II J Func Anal

Ho R Howe Dual pairs in physics Harmonic oscil lators photons

electrons and singletons in Applications of Group Theory in

Phys ics and Mathematical Phys ics M Flato P Sally and G

Zuckermann e ds Lecture s in Applie d Mathematics Vol

Amer Math So c Providence RI

K W H Klink Scattering operators on Fock space IV J Phys A

Math Gen

K Scattering operators on Fock space V J Phys A

Math Gen

KT W H Klink and T TonThat On resolving the multiplicity of

arbitrary tensor products of the U N groups J Phys A Math

Gen

nFold tensor products of GLN C and decomposi KT

tion of Fock spaces J Func Anal

W H Klink and Tuong TonThat

KT Calculation of ClebschGordan and Racah coe

cients using symbolic manipulation programs J Comput Phys

Invariant theory of the block diagonal subgroups of KT

GLn C and generalized Casimir operators J of Alg No

KV M Kashiwara and M Vergne On the SegalShaleWeil representa

tions and harmonic polynomials Inv Math

MQ M Moshinsky and C Que sne Linear canonical transformations

and their unitary representations J Math Phys

Pr C Pro ce s i The invariant theory of n n matrices Adv in Math

Sa M Saito Representations unitaires des groups symplectiques J

Math So c Jpn

Sc I Schur di ss ertation Berlin

We H Weyl The classical groups their invariants and representations

Pr inceton Univers ity Pre ss Pr inceton NJ

c

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