where S k (V ) denotes a homogeneous component of degree k in S(V ). For ν = (ν 1 ; ;ν n ) 2 (Z 0 ) n, put Then we have S ν = S ν1 S νn ρ S jνj (jνj =

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1 Restriction of the irreducible representations of GL n to the symmetric group S n. Kyo Nishiyama Division of athematics, Faculty of Integrated Human Studies, Kyoto University, Kyoto , JAPAN kyo@math.h.kyoto-u.ac.jp 1 Problem Let =( 1 ; ; n ) be a dominant integral weight ofgl n and consider the finite dimensional irreducible representation ρ with the highest weight. The permutation matrices in GL n form a finite subgroup which is isomorphic to the symmetric group S n. We identify S n with this subgroup. Then our problem can be stated as follows. Problem 1.1 Describe the decomposition of ρ j when restricted to the subgroup S n. We will reduce this problem to the decomposition of plethysms in principle. 2 ain result Let us review GL n GL m -duality on the symmetric algebra S(C n ΩC m ). It is well-known that S(C n Ω C m ) is multiplicity free as a representation of GL n GL m and decomposes as follows (see, e.g. [Howe]): S(C n Ω C m ) length()»minfm;ng Ω ; where =( 1 ; 2 ; )isaweight (or a partition) and is the irreducible representation of GL n with highest weight. Put V = C m and assume m n. Then we can reformulate the left hand side of the above formula: S(C n Ω V ) Ω n S(V ) ψ 1 = Ω n S k (V ) k=0 =! ν=(ν 1 ; ;ν n)2z n 0 S ν1 (V ) Ω ΩS νn (V ); 1

2 where S k (V ) denotes a homogeneous component of degree k in S(V ). For ν = (ν 1 ; ;ν n ) 2 (Z 0 ) n, put Then we have S ν = S ν1 S νn ρ S jνj (jνj = ν ν n ): S ν1 (V ) Ω ΩS νn (V )= Ω jνj V S ν : (2.1) By the classical Schur duality (cf. [Weyl]), we know the decomposition jj=jνj;length()»n Ω jνj V jj=jνj;length()»n Ω ff as a GL m S jνj -module. Therefore the above formula (2.1) becomes 0 1 Ω ff AS = Ω (ff ) Sν : We summarize as Ω m S(V ) ν2z n 0 jj=jνj;length()»n length()»n Ω jj=jνj;length()»n Ω (ff ) Sν 1 (ff ) A Sν : ν2z n 0 ;jνj=jj Here, (ff ) Sν becomes an S n -module, whose module structure is induced by the original action of GL n. So we obtain fi ν2z n 0 ;jνj=jj (ff ) Sν μ`jj;length(μ)»n ψ (ff ) Sν! Let V be a representation space on which GL n acts via and V (μ) its weight spaceof weight μ. We put V (S n μ) = V (ν). Then, clearly V (S n μ) is invariant under S n and we get the following lemma. Lemma 2.1 As a representation of S n, there is an isomorphism: V (S n μ) (ff ) Sν : : 2

3 Take a partition μ `jj. Consider the normalizer N μ = NS jμj (S μ ) of S μ in S jμj ; N μ = fs 2 S jμj j ss μ s 1 = S μ g: Then there exists a partition ff(μ) =ff =(ff 1 ; ;ff k )oflength(μ)» n such that N μ =S μ S ff = S ff1 S ffk : Since N μ normalizes S μ, it acts on (ff ) Sμ. oreover, by definition, S μ acts on (ff ) Sμ trivially. So we get a representation of S ff N μ =S μ on (ff ) Sμ. Proposition 2.2 With the notations above, we have μ2s n μ (ff ) Sμ Ind S ff S njffj (ff ) Sμ Ω 1; where 1 means the trivial representation of S njffj. Proof. Now we summarize the above results into Theorem 2.3 fi Ind S ff S njffj (ff ) Sμ Ω 1; μ`jj where ff ` length(μ) is determined by μ via NS jj (S μ )=S μ S ff : Remark. Put k = jj = jμj. By the Frobenius reciprocity, we have (ff ) Sμ HomS μ (C ;ff ) HomS k (Ind S k S μ C ;ff ); =) dim(ff ) Sμ =[ff :Ind S k S μ C ]: This number is called the Kostka number (cf. [acdonald]). By Lemma 2.1, it is equal to the weight multiplicity ofμ in the representation ( ; V )ofgl n. 3

4 3 Relation to the plethysm It seems difficult to determine the action of S ff on (ff ) Sμ. The reason why it is difficult is as follows. Put k = jj (i.e., ` k). Note that the normalizer of S μ = S μ1 S μn in S k is a wreath product S ff n S μ ρ S k ; S ff = S ff1 S ffm for some ff ` n. The irreducible representations of a wreath product are well-studied and completely classified (see [JK, Theorem ]). In our case, for ß 2 S^ff, let(ß : 1) be an irreducible representation of S ff n S μ which is obtained by the successive application of the natural projection of S ff n S μ! S ff and the representation ß of S ff. Then we get HomS ff (ß; (ff ) Sμ ) = HomS ff ((ß :1) Sμ ; (ff ) Sμ ) = Hom ((ß :1);ff ) HomS k (Ind S k (ß :1);ff ): If we know (ff ) Sμ completely as an S ff -module, then we know HomS ff (ß; (ff ) Sμ ), hence the multiplicity of ff in Ind k (ß : 1) can be determined. However, the induced representation Ind S k (ß :1) is a special case of plethysms and its decomposition is not completely known yet (cf. [JK, x5.4]). Using the above relation, we can rephrase our theorem in terms of the decomposition of plethysm: (ff ) Sμ [ff :Ind k (ß :1)]ß: Theorem 3.1 fi μ`jj ß2S^ff ß2S^ff [ff :Ind S k (ß :1)]Ind S ff S njffj ß Ω 1: Example 3.2 Take =(k n )=(k; ;k). In this case, we have =(det (n) ) k. So, we know fi = (sgn) k. oreover, by Lemma 2.1, we get (ff ) Sμ = ρ C if μ =, 0 otherwise. For μ =, the normalizer of S = S k S k in S nk is isomorphic to S n n (S k ) n. So we have ρ [ff (k n ); Ind S nk 1 if ß = (sgn)k, S nn(s k ) n (ß : 1)] = 0 if ß 6= (sgn) k. 4

5 Example 3.3 jj = n and μ =(1; ; 1)=(1 n ). Then ff(μ) =(n) andv(μ) ff. This is a well-known result (cf. [Kostant]). Example 3.4 jj = kn and μ =(k; ;k)=(k n ). Then ff(μ) =(n) and V(μ) ß2 b S n hff :Ind S nk S nn(s k ) n (ß :1) i ß This is a result of [AT]. References [AT] S. Ariki, J. atsuzawa and I. Terada, Representations of Weyl groups on zero weight spaces. In Algebraic and Topological Theories, to the memory of Dr. Takehiko iyata", Kinokuniya, 1985, pp [Gutkin] E. Gutkin, Representations of the Weyl group in the space of vectors of zero weight. Uspehiat. Nauka, 28(1973), (in Russian). [Kostant] B. Kostant, On acdonalds -function formula, the Laplacian and generalized exponents. Adv. in ath., 20(????), [King] R. C. King, Branching rules for GL(N) ff ± m and the evaluation of inner plethysms, J. ath. Phys., 15(1974). [Howe] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. In The Schur lectures (1992)", Israel athematical Conference Proceedings, 8(1995), Bar-Ilan Univ., pp [JK] G. James and A. Kerber, The representation theory of the symmetric group. Encyclopedia of ath. and App. 16, Addison-Wesley, [acdonald] I. acdonald, Symmetric functions and Hall polynomials. Clarendon Press, [Weyl] H. Weyl, The Classical Groups. Princeton University Press, Princeton, N.J., First Draft: 1996/10/2 Ver 1.0 : 1996/10/17, Ver 1.1 : 1996/11/5, Ver. 1.2 [00/11/28 10:11] This version is compiled on November 29,