JOURNAL OF ALGEBRA 195, 233᎐240Ž. 1997 ARTICLE NO. JA977049
Stability of Schur Functors*
Alexandre I. Kabanov†
Department of Mathematics, Michigan State Uni¨ersity, Wells Hall, East Lansing, Michigan 48824-1027
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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 ᒐᒍŽ.ŽV resp. ᒐᒌ 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. ᮊ 1997 Academic Press
INTRODUCTION
Con¨ention. All vector spaces, Lie algebras, and their representations Ž are considered over the complex numbers. Given a partition s 1 G . 2 G иии G k ) 0 we say that k is the length of , and we denote it by ␦Ž. Ýk << . We call is1 i the weight of , and we denote it by . We also say that is a partition of << Žcf.wx 3, I.1. . Each partition determines a Schur functor ޓ which associates to a vector space V an irreducible ᒐᒉŽ.V -module ޓV, and each irreducible representation of ᒐᒉŽ.V can be obtained in this way. This allows us to define a group homomorphism of representation rings : R ᒐᒉ R ᒐᒉ Ž.nn12¨ Ž. whenever n n by sending the representation of ᒐᒉ corresponding to 12F n1 a partition to the representation of ᒐᒉ corresponding to the same n2
* Research supported in part by an Alfred P. Sloan Doctoral Dissertation Fellowship. † On leave until September 1997, Max-Planck-Institut fur¨ Mathematik, Gottfried-Claren- Strasse 26, 53225, Bonn, Germany. E-mail address: [email protected].
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0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 234 ALEXANDRE I. KABANOV partition. Using one can compare the decomposition into irreducible components in RŽ.ᒐᒉn for different n. A functor F which associates to a vector space V a representation FVŽ. of ᒐᒉŽ.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. ᒐᒉ V12[ᒐᒉ V . The stability of the functors ޓޓV and ޓ V12[ 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 ޓ² : associates to V an irreducible representation of ᒐᒍŽ.V . As in the case of ᒐᒉŽ.V this allows us to compare the decomposition into irreducible components in RŽ.ᒐᒍn for different n.
THEOREM Ž.cf. Theorem 7 . If 2<<␦Ž. Fdim V, then the decomposi- tion of the ᒐᒍŽ.V -module ޓޓ Ž² :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 ޓ²:V12[ Vasa Ž. Ž. Ž. ᒐᒍ V12[ᒐᒍ V -module does not depend on dim Vi, i s 1, 2, when 2␦ Ä4 Ž. Fmin dim Vi, and it does not depend on dim V1 pro¨ided that 2␦ 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 ޓ non-degenerate symmetric bilinear forms, and w x be the orthogonal Schur functor associated to .
THEOREM. If 2<<␦Ž. Fdim W, then the decomposition of the Ž.ޓޓ Ž . ᒐᒌ W -module w xW into irreducible components is independent of dim W.
HEOREM ޓ Ž. Ž. T. The decomposition of wx W12[ Wasaᒐᒌ W 1[ Ž. Ž. ᒐᒌ W2 -module does not depend on dim Wi, i s 1, 2, pro¨ided 2␦ F Ä4 Ž. min dim Vi, and it does not depend on dim W1 pro¨ided that 2␦ 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 of d one can define the corresponding Young
symmetrizer c lying in the group ring of the symmetric group ⌺d wx1, p. 46 . The Schur functor ޓ takes a vector space V to the image of c:
md md ޓV s ImŽ.c: V ª V .
Let n be the dimension of V. Then the ᒐᒉn-modules ޓV with ␦Ž.-n are in one-to-one correspondence with the equivalence classes of irre-
ducible representations of ᒐᒉn wx1, Proposition 15.15 . We denote by Ä4e , ␦Ž.-n, the corresponding additive basis of RŽᒐᒉ ..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 F if iiF for all i, and - if in addition ii-for 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 <<␦Ž. -n,then
ޓŽ.es Ý Ce,1.1Ž. :␦Ž.F<<␦Ž.
where C are integers independent of n.
Fix an inclusion ᒐᒉlm[ᒐᒉ ¨ᒐᒉ lqm. It induces the restriction homo- morphism : RŽ.ŽᒐᒉlqmlmªR ᒐᒉ [ᒐᒉ . 236 ALEXANDRE I. KABANOV
XY commuting with the Schur functors. Let Äee 4,␦Ž.-l,␦ Ž .-m,bean Ž. additive basis of R ᒐᒉlm[ᒐᒉ whose elements correspond to irreducible representations.
PROPOSITION 21,p.80.wxIf ␦Ž.-minÄ4l, m , then
X Y XY Ž.1.2 Ž.es ޓ ŽeŽ1.Žq e 1. .s Ý Me e , , : F , F
where M are Littlewood᎐Richardson coefficients and they do not depend on l, m. If m is fixed and ␦Ž.Fl,then
X Y X X Y Ž.1.3 Ž.es ޓ ŽeŽ1.Žq e 1. .s Ý Me e , ,:F,␦Ž.-m
X where M are integers independent of l.
Another ingredient we will need is the transition from the basis Ä4e of RŽŽ..ᒐᒉ V to the basis corresponding to the products of the symmetric powers. We denote by SymV the tensor product
Sym 1V m Sym 2 V m иии m Sym k V,
where is a partition Ž.12, ,...,k. We also introduce a lexicographic ordering on the set of partitions of the
same weight: we say that $ if the first non-vanishing iiy is positive.
PROPOSITION 3wx 1, p. 86; 3, I.6.5 . There is an isomorphism of ᒐᒉŽ.V - modules
SymV ޓ V K ޓ V , ( [ ž/[ :<<<<s,$
where K are non-negati¨e integers independent of dim V.
This implies that the transition matrix Ž.K 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 ޓ² : associates to a pair Ž²:.V, , the ᒐᒍ2 p-module
² d: ² d: ² d: ޓ²:V s V l ޓV s ImŽ.c : V ª V .
The symplectic Schur functor ޓ²:V vanishes if and only if ␦Ž.)p. There is a one-to-one correspondence between irreducible representations of ᒐᒍ2 p and partitions whose length is bounded by p wx1, Theorem 17.11 . Ä4 Ž. Let f , ␦Fp, be an additive basis of the representation ring RŽ.ᒐᒍ2 p whose generator f corresponds to the irreducible representation ޓ² :V. The following proposition shows that the multiplication in RŽ.ᒐᒍ2 p is stable. The restrictions on the index of summation come from properties of Littlewood᎐Richardson numbers.Ž These are the integers M from Proposition 2..
PROPOSITION 5wx 1, p. 424; 4, Sect. 4 . If ␦Ž.q␦ Ž .Fn,then
ffs Ý Nf, :<<F <<q < <,␦Ž.F␦ Ž.q␦ Ž . where N are non-negati¨e integers independent of n.
An inclusion of Lie algebras ᒐᒍ2 p ¨ᒐᒉ2 p induces the restriction homo- s Ž. Ž. morphism : R ᒐᒉ2 p ªR ᒐᒍ2 p of their representation rings. PROPOSITION 6wx 1, p. 427; 4, Sect. 5 . If ␦Ž.Fp,then
s Ž.es fq ÝLf, - where L are non-negati¨e integers independent of n.
3. PLETHYSM OF SYMPLECTIC SCHUR FUNCTORS
In this section we prove that the composition of a symplectic Schur functor and a Schur functor decomposes stably as a representation of the symplectic group. We keep the notation of the previous sections.
THEOREM 7. If <<␦Ž. Fp,then
ޓŽ.fs Ý Pf, :␦Ž.F<<␦Ž. where P are integers independent of p. 238 ALEXANDRE I. KABANOV
Proof. First we prove by induction a special case of the theorem when Ž. aŽ. safor some integer a. That is, we want to show that Sym f s Ž. Ž. ÝPfŽa., where the index varies over partitions with ␦ Fa␦, and the integers PŽa. are independent of p. The assertion is clearly satisfied for all pairs Ž.a, where a s 0, 1. We Ž. assume that the assertion is true for all pairs a11, such that a1F a, 1 F, and at least one of these inequalities is strict, and prove that the assertion is also true for the pair Ž.a, . s Ž. Ž. We apply the restriction homomorphism : R ᒐᒉ2 p ªR ᒐᒍ2 p to the equalityŽ. 1.1 . Using Proposition 6 one gets that
a Sym fq ÝÝÝLf s CLfŽa.. ž/11 ž/11 1- :␦Ž.Fa␦ Ž. 1F
The coefficients of the right hand side do not depend on p as ␦Ž.F a␦Ž.Fp. The left hand side can be written as
a aayii Sym Ž.fq ÝÝSym Ž.fSym Lf. ž/11 is1 1-
By the induction hypothesis and Proposition 5 the coefficients of
ayi i Sym Ž.f and Sym ÝLf ž/11 1-
a do not depend on p. Thus we conclude that the coefficients of Sym Ž.f , i.e., the integers PŽa., do not depend on p. The property P Ža./ 0 only if ␦Ž.Fa␦ Ž.is clear. Taking the tensor products of symmetric powers and using Proposition 5 again, one immediately derives that if <<␦Ž. Fp, then
X Sym Ž.fs Ý Pf , :␦Ž.F<<␦Ž.
X where the integers P are independent of p. The last step is to relate the symmetric functors and the Schur functors 1 using Proposition 3. One can express ޓŽ.fthrough Sym Ž.f such that <<<< because the transition matrix Ž.K is unitriangular. This 1 s 1 implies that the integers P are independent of p and vanish when ␦Ž.)<<␦Ž. . STABILITY OF SCHUR FUNCTORS 239
4. RESTRICTION OF SYMPLECTIC SCHUR FUNCTORS
Let V12and V be two complex vector spaces of dimensions 2l and 2m, respectively, with non-degenerate skew-symmetric forms²: ,12 and ²: , . ²: ²: ²: Then the form , s , 12q ,onV12[Vis also non-degenerate and skew-symmetric. The direct sum determines an inclusion ᒐᒍ2l [ᒐᒍ2m ;ᒐᒍ2 p, where p s l q m, and therefore the restriction homomorphism
s : RŽ.ᒐᒍ2pªR Žᒐᒍ2l[ᒐᒍ2m .. Ž. ÄXY4 Ž. Ž . The ring R ᒐᒍ2l[ᒐᒍ2m has an additive basis ff,␦-l,␦Fm, X Y such that the element f f corresponds to the irreducible representation ޓ² :V1 G ޓ² :V2 . THEOREM 8. If ␦Ž.FminÄ4l, m , then
s XY Ž.fs Ý Qf f , , : , F where the Q are integers independent of l, m. If m is fixed, and ␦Ž.Fl,then
s X X Y Ž.fs Ý Qf f , ,:F,␦Ž.Fm
X where the Q are integers independent of l. Proof. We will treat both cases simultaneously, and prove the theorem by induction on . The assertion is clear when is empty of s Ž.1.We assume that the statement of the theorem is true for all Ј - , and prove the theorem for Ј s . Apply the restriction homomorphism
s s m : RŽ.Ž.ᒐᒉ2l[ᒐᒉ2mªR ᒐᒍ2l[ᒐᒍ2m toŽ. 1.2 and Ž. 1.3 . Using Proposition 6 one gets
s s X Y Ž.fq ÝÝÝÝL Ž.f s MLf Lf 11 ž/11ž/ 11 1-,F1F 1F in case ␦Ž.FminÄ4l, m , and
s s X X X Y Ž.fq ÝÝÝÝL Ž.f s MLf Lf 11 ž/11ž/11 1- F,1F ␦Ž.1Fm in case m is fixed, and ␦Ž.Fl. 240 ALEXANDRE I. KABANOV
The coefficients of the right hand side are independent of l, m Ž.resp. l Ž. Ž. Ž . Ž. as ␦1 F␦ Fl, and in the first case in addition ␦1 F␦ Fm. The coefficients of Ý L sŽ.f are independent of l, m Ž.resp. l by 11- 1 s the induction hypothesis. This implies that the coefficients of Ž.f , that X is, the integers Q Žresp. Q .Ž, are independent of l, m resp. l.. The conditions on the indices , are clear.
ACKNOWLEDGMENTS
I am grateful to R. Hain who motivated and encouraged me in writing this paper, and with whom I had many useful conversations. I thank W. Fulton and E. Getzler for their discussions, and the referee for suggestions in improving the exposition. I also thank the Max-Planck-Institut fur¨ Mathematik for its hospitality where the final version was written.
REFERENCES
1. W. Fulton and J. Harris, Representation theory, a first course, in ‘‘Graduate Texts in Math.,’’ Vol. 129, Springer-Verlag, New YorkrBerlin, 1991. 2. R. Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc., in press. 3. I. G. Macdonald, ‘‘Symmetric Functions and Hall Polynomials,’’ Clarendon, Oxford, 1979, 1995. 4. R. C. King, Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups, J. Math. Phys. 12 Ž.1971 , 1588᎐1598.