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Duality in Representation Theory 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
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