JOURNAL OF ALGEBRA 195, 233]240Ž. 1997 ARTICLE NO. JA977049 Stability of Schur Functors* Alexandre I. Kabanov² Department of Mathematics, Michigan State Uni¨ersity, Wells Hall, East Lansing, Michigan 48824-1027 View metadata, citation and similar papersCommunicated at core.ac.uk by Peter Littelmann brought to you by CORE provided by Elsevier - Publisher Connector Received September 5, 1996 SymplecticŽ. resp. orthogonal Schur functors are defined on a vector space V with a non-degenerate skew-symmetricŽ. resp. symmetric bilinear form. Each of these functors determines an irreducible representation of spŽ.ŽV resp. so Ž..V . We prove that if dim V is sufficiently large, then the splitting into irreducible components of the composition of a symplecticŽ. orthogonal Schur functor and an ordinary Schur functor applied to V is independent of dim V. We also prove the stability of the branching for the symplecticŽ. orthogonal Schur functors. Q 1997 Academic Press INTRODUCTION Con¨ention. All vector spaces, Lie algebras, and their representations Ž are considered over the complex numbers. Given a partition l s l1 G . l2 G ??? G lk ) 0 we say that k is the length of l, and we denote it by dlŽ. Ýk l l <<l . We call is1 i the weight of , and we denote it by . We also say that l is a partition of <<l Žcf.wx 3, I.1. Each partition l determines a Schur functor Sl which associates to a vector space V an irreducible slŽ.V -module SlV, and each irreducible representation of slŽ.V can be obtained in this way. This allows us to define a group homomorphism of representation rings t : R sl R sl Ž.nn12¨ Ž. whenever n n by sending the representation of sl corresponding to 12F n1 a partition l to the representation of sl corresponding to the same n2 * Research supported in part by an Alfred P. Sloan Doctoral Dissertation Fellowship. ² On leave until September 1997, Max-Planck-Institut furÈ Mathematik, Gottfried-Claren- Strasse 26, 53225, Bonn, Germany. E-mail address: [email protected]. 233 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved. 234 ALEXANDRE I. KABANOV partition. Using t one can compare the decomposition into irreducible components in RŽ.sln for different n. A functor F which associates to a vector space V a representation FVŽ. of slŽ.V is called stable if the decomposition of FV Ž.into irreducible components is independent of dim V provided that dim V is sufficiently Ž large. One can similarly define stability with respect to V12[ V and Ž. Ž.. Ž.Ž. sl V12[sl V . The stability of the functors SSmlV and S nV12[ V is a classical resultwx 1 . In this paper we prove similar stability results for the symplectic and orthogonal Schur functors. We first state the results in the symplectic case. Let V be a vector space with a non-degenerate skew-symmetric bilinear form. A symplectic Schur functor S² l: associates to V an irreducible representation of spŽ.V . As in the case of slŽ.V this allows us to compare the decomposition into irreducible components in RŽ.spn for different n. THEOREM Ž.cf. Theorem 7 . If 2<<mdŽ. l Fdim V, then the decomposi- tion of the spŽ.V -module SSm Ž² l:V . into irreducible components is indepen- dent of dim V. Let Vi, i s 1, 2, be vector spaces with non-degenerate skew-symmetric bilinear forms. Ž. Ž. THEOREM cf. Theorem 8 . The decomposition of S²n:V12[ Vasa Ž. Ž. Ž. sp V12[sp V -module does not depend on dim Vi, i s 1, 2, when 2dn Ä4 Ž. Fmin dim Vi, and it does not depend on dim V1 pro¨ided that 2dn F dim V12 when dim V is fixed. Now we state the results in the orthogonal case. They are identical to those in the symplectic case. Let W, W12, and W be vector spaces with S non-degenerate symmetric bilinear forms, and w lx be the orthogonal Schur functor associated to l. THEOREM. If 2<<mdŽ. l Fdim W, then the decomposition of the Ž.SS Ž . so W -module m w lxW into irreducible components is independent of dim W. HEOREM S Ž. Ž. T. The decomposition of wnx W12[ Wasaso W 1[ Ž. Ž. so W2 -module does not depend on dim Wi, i s 1, 2, pro¨ided 2dn F Ä4 Ž. min dim Vi, and it does not depend on dim W1 pro¨ided that 2dn F dim W12 when dim W is fixed. The only difference between the symplectic and the orthogonal cases is that in the latter case the orthogonal Schur functors describe a subring of the representation ring rather than the whole ring. However, all represen- tations which we consider in this note can be decomposed into irreducible STABILITY OF SCHUR FUNCTORS 235 components inside these subrings, and the difference above creates no difficulties. The proofs in the orthogonal case are identical to those in the symplectic case, and therefore will be omitted. The theorems above are a part of the mathematical folklore, but I am not aware of a relevant reference. The primary motivation for writing this paper is an article of Richard Hainwx 2, Sect. 6 . He uses the stability of the composition for the symplectic Schur functors in the study of the Torelli Lie algebras. This paper has the following structure. In the first two sections we recall relevant facts concerning stability in the representation rings of the special linear and symplectic Lie algebras. In the next two sections we use these facts to prove the stability of the compositionŽ. plethysm and branching for the symplectic Schur functors. 1. CLASSICAL SCHUR FUNCTORS For each partition l of d one can define the corresponding Young symmetrizer cl lying in the group ring of the symmetric group Sd wx1, p. 46 . The Schur functor Sl takes a vector space V to the image of cl: md md SlV s ImŽ.cl: V ª V . Let n be the dimension of V. Then the sln-modules SlV with dlŽ.-n are in one-to-one correspondence with the equivalence classes of irre- ducible representations of sln wx1, Proposition 15.15 . We denote by Ä4el , dlŽ.-n, the corresponding additive basis of RŽsl ..n . Now we list several stability properties of this representation ring. In order to do this we need to introduce a partial order on the set of all partitions. We say that m F l if miiF l for all i, and m - l if in addition mii-lfor some i. We also adopt a convention that the value of a functor on an element of a representation ring is determined by the value of this functor on the corresponding representation. PROPOSITION 1wx 1, Example 6.17; 3, I.8 . If <<mdŽ. l -n,then SmŽ.ells Ý Cemnn,1.1Ž. n:dnŽ.F<<mdŽ. l where Clmn are integers independent of n. Fix an inclusion sllm[sl ¨sl lqm. It induces the restriction homo- morphism r : RŽ.ŽsllqmlmªR sl [sl . 236 ALEXANDRE I. KABANOV XY commuting with the Schur functors. Let Äeel m4,dlŽ.-l,dm Ž .-m,bean Ž. additive basis of R sllm[sl whose elements correspond to irreducible representations. PROPOSITION 21,p.80.wxIf dnŽ.-minÄ4l, m , then X Y XY Ž.1.2 r Ž.ens Sn ŽeŽ1.Žq e 1. .s Ý Melmn le m , l, m: lFn , mFn where Mlmn are Littlewood]Richardson coefficients and they do not depend on l, m. If m is fixed and dnŽ.Fl,then X Y X X Y Ž.1.3 r Ž.ens Sn ŽeŽ1.Žq e 1. .s Ý Mlmne le m , l,m:lFn,dmŽ.-m X where Mlmn are integers independent of l. Another ingredient we will need is the transition from the basis Ä4el of RŽŽ..sl V to the basis corresponding to the products of the symmetric powers. We denote by SymlV the tensor product l l l Sym 1V m Sym 2 V m ??? m Sym k V, where l is a partition Ž.l12, l ,...,lk. We also introduce a lexicographic ordering on the set of partitions of the same weight: we say that m $ l if the first non-vanishing liiy m is positive. PROPOSITION 3wx 1, p. 86; 3, I.6.5 . There is an isomorphism of slŽ.V - modules SymlV S V K S V , ( l [ ž/[ ml m m:<<<<msl,m$l where Kml are non-negati¨e integers independent of dim V. This implies that the transition matrix Ž.Kml is unitriangular, and therefore invertible over the integers. 2. SYMPLECTIC SCHUR FUNCTORS Suppose that a vector space V of dimension n s 2 p carries a non- degenerate skew-symmetric form²: , . This form determines a contrac- d Žd 2. tion V m ª V m y for each two distinct factors of the tensor product. ² d: d Let V : V m be the intersection of the kernels of all these contrac- tions. STABILITY OF SCHUR FUNCTORS 237 DEFINITION 4wx 1, pp. 263]264 . The symplectic Schur functor S² l: associates to a pair Ž²:.V, , the sp2 p-module ² d: ² d: ² d: S²l:V s V l SllV s ImŽ.c : V ª V . The symplectic Schur functor S²l:V vanishes if and only if dlŽ.)p. There is a one-to-one correspondence between irreducible representations of sp2 p and partitions whose length is bounded by p wx1, Theorem 17.11 .