Minimal Free Resolutions of Determinantal Ideals and Irreducible Representations of the Lie Superalgebra gl(m | n)

197, 559]583 Ž1997. JA977101

JOURNAL OF ALGEBRA ARTICLE NO.

Minimal Free Resolutions of Determinantal Ideals and Irreducible Representations of the Lie Superalgebra glŽ m ¬ n. Kaan Akin Department of Mathematics, Uni¨ ersity of Oklahoma, Norman, Oklahoma 73019

and Jerzy WeymanU Department of Mathematics, Northeastern Uni¨ ersity, Boston, Massachusetts 02155 Communicated by D. A. Buchsbaum Received September 27, 1993

1. INTRODUCTION Let K be a field of characteristic zero and let Mm= nŽ K . denote the space of m = n matrices with entries in K. The coordinate ring of Mm= nŽ K . is the polynomial algebra SsK

 x i j ¬ 1 F i F m, 1 F j F n4

Ž 1.

over K in the standard coordinate functions x i j on Mm=nŽ K .. The homogeneous ideal of S generated by the p = p minors of the m = n matrix Ž x i j . is a prime ideal, denoted I p , whose zero set is the determinantal subvariety of Mm= nŽ K . consisting of matrices of rank less than p. This subvariety is preserved under the natural action of the algebraic group GL mŽ K . = GL nŽ K . on the affine space Mm=nŽ K . by row and column operations, and the ideal I p is preserved under the corresponding group action on the coordinate algebra S. The only prime ideals of S that are invariant under the action of GL mŽ K . = GL nŽ K . are the ideals I p for * Partially supported by NSF and RSDF grants. 559 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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p G 1. The purpose of this paper is to use minimal S-free resolutions of the ideals I p to construct some atypical irreducible representations of the Lie superalgebra slŽ m ¬ n.K from the Koszul duals of the ideals I p . In order to make the group actions more explicit, we will identify the space Mm= nŽ K . with Hom K Ž G, F ., where F and G are K-vector spaces of dimensions m and n, respectively. Under the natural identification of Hom K Ž G, F . with F mK GU , where GU denotes the dual space Hom K Ž G, K ., we can identify the polynomial algebra S with the symmetric algebra S Ž F U mK G . on the dual space F U mK G of F mK GU . The standard action of the group GLŽ F . = GLŽ G . on the vector space F U m G extends naturally to an action on the graded algebra SŽ F U m G .. The Cauchy formula Si Ž F U mK G . s

Ý Ll F U mK

l&i

Ll G

Ž 2.

gives a direct sum decomposition of the graded component S i into irreducible GLŽ F . = GLŽ G .-modules,1 where the sum is taken over partitions l of i with terms vanishing unless l1 F minŽ m, n.. When l is the partition Ž p ., the corresponding term Ll F U m Ll G is the tensor product L p F U m L p G of pth exterior powers and this term generates the determinantal ideals I p . More generally, we let Il denote the homogeneous ideal of S generated by Ll F U mK Ll G. A detailed discussion of the ideals Il , the minimal GLŽ F . = GLŽ G .-invariant ideals of S, can be found in w7x. In w16x, Lascoux gave an explicit description of the chains in the graded minimal S-free resolution of the quotient SrI p in terms of irreducible representations of the algebraic group GLŽ F . = GLŽ G .. Following the notation of w21x, the Lascoux resolution is the total complex of a chain complex d lp

d kp

d1p

p 0 ª Wlp ª ??? ª Wkp ª Wky1 ª ??? ª W1p ª S ª 0

Ž 3.

of chain complexes Wkp of graded free S-modules, where l s minŽ m, n. y p q 1, and the unaugmented Lascoux complex with the initial S deleted is a resolution of I p . The module ŽWkp . i of chains in homological degree i is the direct sum

Ý S mK

a, b

U & && LŽ a 1qt , . . . , a kqt , & b 1 , b 2 , . . . . F mK LŽ b 1 qt , . . . , b k qt , a 1 , a 2 , . . . . G

Ž 4.

over all pairs a , b of partitions a s Ž a 1 , . . . , a k ., b s Ž b 1 , . . . , b k . whose total weight a 1 q ??? qa k q b 1 q ??? qb k equals i and where t s p q See the beginning of Section 3 for a description of the irreducible Schur modules LlŽ F U . and LlŽ G .. 1

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p k y 1. The differentials d kp : Wkp ª Wky1 are homogeneous maps of bidegree Ž p, 2 k y 2., where p is the degree over S and 2 k y 2 is the homological degree; the internal differential of the chain complex Wkp is of bidegree Ž1, y1.. Consequently, the term ŽWkp . i appears in homological degree k 2 q i of the resolution and is generated as a graded free S-module by homogeneous generators of degree k Ž p q k y 1. q i. The interpretation of the complexes Wkp in the context of Koszul duality is given in Section 2. It was shown in w21x that each component complex Wkp can be constructed as the homology of a double complex L?k, p of chain complexes built out of certain Schur complexes. Let X kp denote the reduction K mS Wpk of the complex Wkp modulo the homogenous maximal ideal Sqs I1. The group GLŽ F . = GLŽ G . acts on the graded vector space X kp s Ý i Ž X kp . i and the homogeneous component Ž X kp . i is the direct sum

Ý LŽ a qt , . . . , a qt , b˜ , b˜ , . . . . F U mK LŽ b qt , . . . , b qt , a˜ , a˜ , . . . .G,

a, b

1

k

1

2

1

k

1

2

Ž 4.

X

where a , b are as in Ž4.. In Section 3 we utilize the construction of Wkp from L?k,? p to show that the action of the Lie algebra glŽ F . = glŽ G . on the graded vector space X kp extends to an action of the Lie superalgebra glŽ m ¬ n. s glŽ F ¬ G . of endomorphisms of the Z 2-graded vector space F [ G with even component F and odd component G. The self-duality of U the double complex L?k,? p implies that the glŽ m ¬ n.-module X kp is self-dual with respect to an appropriate weight-preserving duality functor.2 On the other hand, a comparison of Roberts’ minimal complex with Lascoux’s U resolutions is used in the proof of Theorem 4.3 to showU that X kp is a highest weight module for glŽ m ¬ n.. The irreducibility of X kp over glŽ m ¬ n., or equivalently over slŽ m ¬ n., then follows because any self-dual highest weight module must be irreducible. Since the formal character of a Schur module LlŽ E . of a vector space E of dimension r is the classical Schur function slŽ x 1 , . . . , x r . in r variables, the decomposition in Ž4.X gives a character for the GLŽ F U . = GLŽ G .module X kp as a sum of products of Schur functions in two sets of variables, in the same way that the decomposition in Ž2. corresponds to the classical form of Cauchy’s formula in the setting of symmetric functions. In a category of representations such as finite dimensional glŽ m ¬ n.modules, it is natural to look for character formulas for simple modules in terms of universal highest modules. One can also try to construct resolutions realizing character formulas of this type, as for example the Bernstein]Gelfand]Gelfand resolution realizes the Weyl character formula. The duality functor Žy.U t is described after the general discussion of linear duals of Lie supermodules in the beginning of Section 4. 2

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The universal highest weight modules in the category of finite dimensional glŽ m ¬ n.-modules 3 are the Kac modules V Ž x . defined in w11, 12, 13x and a finite dimensional simple glŽ m ¬ n.-module of highest weight x , denoted V Ž x ., is a quotient of the Kac module V Ž x . of highest weight x Žsee Section 2 for details.. A simple glŽ m ¬ n.-module V Ž x . is called typical if and only if V Ž x . s V Ž x .. The atypicality of the simple modules X kp and pU X k can be seen easily either from their construction out of Lascoux’s resolution or from the criteria on highest weights given in w12, Section 4x Žsee the statement at the beginning of Section 5.. In the case p ) 1 there is a minimal presentation of X 1p over slŽ m ¬ n. by Kac modules, V Ž p q 1, 0, . . . , 0, yp y 1 . [ V Ž p, 1, 0, . . . , 0, y1, yp . ª V Ž p, 0, . . . , 0, yp . ª X 1p ª 0, U

and analogous presentations exist of all X kp and X kp for p ) 1 Žsee Theorem 5.2.. In the case p s 1 the same type of degree 1 relations are still there but they have to be supplemented by higher degree relations. Some of the problems in finding minimal presentations for atypical simple glŽ m ¬ n.-modules and extending them to resolutions, touched on briefly in Section 5, are dealt with in w25x. Techniques of this paper are developed further in w25x and in particular yield character formulas for the modules X kp in terms of Kac modules which confirm in these special cases the general character formulas conjectured in w10x for all atypical slŽ m ¬ n.modules. The problem of finding minimal presentations of the other atypical simple glŽ m ¬ n.-modules and extending presentations to resolutions is discussed in Section 5.

2. KOSZUL DUALITY AND REPRESENTATIONS OF glŽ m ¬ n. Let A be a finitely generated quadratic algebra over a field K of characteristic zero, meaning that A is a graded algebra Ý iG 0 A i generated over A 0 s K by the finite dimensional vector space A1 with relations of degree 2. More explicitly, A is isomorphic to the quotient TK Ž A1 .rŽW . of the tensor algebra TK Ž A1 . over K on the vector space A1 modulo the two sided ideal ŽW . generated by a subspace W of the space T2 Ž A1 . s A1 m A1 of quadratic tensors. The quadratic dual of A is the quadratic algebra A!s Ý i G 0 A!i over K generated by the dual space AU1 s A!1 of A1 with 3

Unless stated explicitly otherwise, ‘‘module’’ is understood to mean ‘‘left-module’’ throughout this paper, and ‘‘action’’ to mean ‘‘left-action.’’

IDEALS AND REPRESENTATIONS OF

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relations W H , that is, Tk Ž AU1 .rŽW H. , where W H is the annihilator W H s  x g AU1 mK AU1 ¬ ² x, W : s 0 4

Ž 6.

of W in AU1 mK AU1 under the natural pairing. Clearly, A!! is naturally isomorphic to A. Now let M s Ý i G q Mi be a graded A-module with a finite free linear presentation A mK Y ª A mK MQ ª M ª 0,

Ž 7.

meaning that M is the quotient of the graded free A-module A m Mq on the finite dimensional vector space Mq modulo the submodule generated by a subspace Y of A1 mK Mq . The linear Koszul dual of M is the graded A!-module M = s Ý i G q Mi= determined by the free linear presentation A !mK Y Hª A !mK Mq= ª M = ª 0,

Ž 8.

where Mq= is the dual space MqU and Y H is the annihilator Y H s  x g AU1 mK MqU ¬ ² x, y : s 0 4

Ž 9.

of Y in AU1 m MqU s A!1 mK Mq=. It is again clear from the finiteness conditions that M = = is naturally isomorphic to M. As in the Introduction, let F and G denote vector spaces of dimensions m and n over a field K of characteristic zero. If we take A to be the symmetric algebra SK Ž F U mK G . over K on the vector space F U mK G, then the quadratic dual A! is naturally isomorphic to the exterior algebra L K Ž F mK GU . on F mK GU . The group GLŽ F . = GLŽ G . acts naturally on the graded algebras A and A! through the standard actions on F U mK G and F mK GU . By an action of GLŽ F . = GLŽ G . on a graded A-module M we mean a homogeneous algebraic action which is compactible with the A-module structure on M in the sense that the module multiplication map A mK M ª M is GLŽ F . = GLŽ G .-equivariant with the group GLŽ F . = GLŽ G . acting diagonally on the tensor product A mK M. Given such an action on a graded A-module M having a finite free linear presentation A mK Y ª A mK X ª M ª 0

Ž 10 .

as in Ž2., where X s Mq and Y is the kernel of the module multiplication map from A1 mK Mq onto Mqq1 , it is clear that X, Y are GLŽ F . = GLŽ G .-modules and that the presentation is a GLŽ F . = GLŽ G .equivariant sequence with the group GLŽ F . = GLŽ G . acting diagonally on the free modules A mK X and A mK Y. It follows easily that the dual

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linear presentation A !mK Y Hª A !mK X U ª M = ª 0

Ž 11 .

is a GLŽ F . = GLŽ G .-equivariant sequence of A!-modules in the same sense. Before explaining how these observations relate to our constructions of glŽ m ¬ n.-modules, we will briefly discuss the representation theory setup for glŽ m ¬ n., which we will make use of throughout the rest of this paper. We let V denote our basic Z 2-graded vector space V0 [ V1 with V0 s F and V1 s G. The superbracket on the Z 2-graded endomorphism algebra Hom K Ž V, V . is defined by the rule

w f , c x s f ( c y Ž y1. degŽ f . degŽ c . c ( f

Ž 12 .

on homogeneous elements. We will denote the resulting Lie superalgebras as glŽ V . or glŽ F ¬ G ., and when working with a specific choice of bases for F and G, also as glŽ m ¬ n.. The Lie superalgebra glŽ V . carries a compatible Z-grading gl Ž V . s gly1 Ž V . m gl 0 Ž V . [ gl 1 Ž V . ,

Ž 13 .

which arises from viewing the grading V0 [ V1 on V as a Z-grading. The subalgebra gl 0 Ž V . of even endomorphisms is naturally isomorphic to the sum glŽ F . [ glŽ G . or ordinary Lie algebras, and the two odd components gl 1 Ž V . s Hom K Ž F , G . ,

gly1 Ž V . s Hom K Ž G, F .

Ž 14 .

may be naturally identified with the spaces F U mK G and F mK GU , respectively. Furthermore, the supertrace of an even endomorphism Ž f , c . g glŽ m. [ glŽ n. is the difference trŽ f . y trŽ c . of ordinary traces, and the supertrace of an odd endomorphism is zero. The supertrace map str: gl Ž m ¬ n . ª K s gl Ž 1 ¬ 0 .

Ž 15 .

is a Lie homomorphism of Z-graded superalgebras and the resulting kernel is the Lie superalgebra slŽ m ¬ n. with the Z-grading restricted from glŽ m ¬ n.. Let g s gy1 [ g 0 [ g 1 denote either of the two Z-graded Lie superalgebras glŽ m ¬ n. or slŽ m ¬ n.. We let p denote the nonnegative subalgebra g 0 [ g 1 and py denote the nonpositive subalgebra. If we let V0 Ž x . denote the simple g 0-module of highest weight x and inflate V0 Ž x . into a p-module through the natural surjection p ª g 0 sending g 1 to zero, then the Kac module V Ž x . of highest weight x can be defined as the g-module Ind gp Ž V0 Ž x .. induced up from p. The g-module V Ž x . s UŽ g . mUŽ p . V0 Ž x . is easily seen to be a finite dimensional highest weight module of highest

IDEALS AND REPRESENTATIONS OF

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565

weight x and hence as a simple top V Ž x .rradŽ V Ž x .. which is denoted by V Ž x .. ŽSee w11, 12, 13x.. As mentioned earlier, one of the equivalent conditions stated in w13, Sect. 4x for the simple g-module V Ž x . to be called typical is that V Ž x . s V Ž x .. By the Poincare]Birkhoff]Witt Theorem for Lie superalgebras, the ´ enveloping algebra UŽ g . of g s gy1 [ p is isomorphic to the tensor product UŽ gy1 . mK UŽ p . as a Žleft gy1 , right p .-bimodule. Consequently, there is a natural isomorphism V Ž x . ( U Ž gy1 . mK V0 Ž x . ,

Ž 16 .

where gy1 acts through the left tensor factor, g 0 acts diagonally on the tensor factors, and g 1 acts in a more complicated manner. Since gy1 is a purely odd abelian Lie superalgebra, its enveloping algebra UŽ gy1 . is simply the exterior algebra LŽ gy1 . on the vector space gy1 s F mK GU , and a Kac module V Ž x . is naturally isomorphic to ! Ž . LŽ F mK GU . mK V0 Ž x . s Am K V0 x as a module over the distinguished y parabolic subalgebra p s gy1 [ g 0 with gy1 acting on the left factor and g 0 acting diagonally. Going back to Koszul duals, recall that if we start with a GLŽ F . = GLŽ G .-equivariant finite free linear presentation A mK Y ª A mK X ª M = ª 0

Ž 17 .

over A s SŽ F U mK G . then the dual linear presentation A !mK Y Hª A !mK X U ª M = ª 0

Ž 18 .

is also GLŽ F . m GLŽ G .-equivariant and hence is seen to be a sequence of py-modules, because A!s UŽ gy1 . and g 0 is the Lie algebra of GLŽ F . = U ! ! H GLŽ G .. Moreover, the free A!-modules Am and Am are K X K Y y isomorphic as p -modules to direct sums of Kac modules. Of obvious interest are the cases where the presentation of M = is actually a sequence of g-modules, some examples of which are discussed in Section 5. As an illustration, take M to be the determinantal ideal I p which has a GLŽ F . = GLŽ G .-equivariant linear presentation A mK

pq1

ž

pq1

L F U mK L p , 1G [ L p , 1 F U m L G p

/

p

ª A mK L F U mK L G ª I p ª 0

Ž 19 .

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and the linear Koszul dual I p= has a presentation A !mK

pq1

ž

pq1

L F mK L GU [ L p , 1 F mK L p , 1GU p

/

p

ª A !mK L F mK L GU ª I p= ª 0

Ž 20 .

of py-modules which is a special case of the g-module presentation described in Theorem 5.2. For p ) 1, the simple g-module X 1p is the K-linear dual I p=* s Hom K Ž I p= , K . of the linear Koszul dual I p= of the ideal I p . In order to explain how the other simple g-modules X kp are likewise related to ‘‘higher’’ Koszul duals of I p , we will sketch contravariant Koszul duality between the bounded derived categories of finitely generated graded modules over the symmetric algebra A s SŽ F U mK G . and over its quadratic dual, the exterior algebra A!s LŽ F mK GU .. We let I denote the minimal injective ‘‘Koszul’’ resolution of the trivial A-module K s ArAq in the category of graded A-modules I is a cochain complex 0 ª I 0 ª I 1 ª I 2 ª ??? ª I i ª I kq 1 ª ???

Ž 21 .

of graded A-modules I k s AUgr mK A!k , where AUgr denotes the graded K-linear dual Ý i AUi s Ý i Hom K Ž A i , K . and the grading I0k s Ý j I jk is defined by I ?k s AUkyj mK A!k . Given any chain complex C ? ? ??? ª Ciq1 ª Ci?ª Ciy1 ª ???

Ž 22 .

of graded A-modules Ci?s Ý j Cij, the contravariant Koszul dual =ŽC. is a chain complex ?

?

?

??? ª Ž C . iq1 ª = Ž C . i ª = Ž C . iy1 ª ???

Ž 23.

of graded A!-modules =Ž C . i?s Ý j=Ž C . ij, where the bigraded component ! =Ž C . ij of =ŽC. s Am C gr* is the vector space t* t s Ý Hom K Ž Ctyi A !jyt . Ý A!jyt m Ctyi

t

Ž 24 .

t

and the differential of =ŽC. is the same as that of the cochain complex Ž . Hom gr A C, I under the natural identification as vector spaces. The relationship between the bigrading on the chain complex =ŽC. and the standard Ž . bigrading on the cochain complex Hom gr A C, I is given under the natural

IDEALS AND REPRESENTATIONS OF

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identification by j

=Ž C . i s

jyi Ý Hom grA Ž C, I. yj

Ž 25 .

and the boundary map of =ŽC. has bidegree Ž y10 . whereas the coboundary Ž . Ž1. map of Hom gr A C, I has bidegree 0 . The restriction of the functor = to bounded chain complexes of finitely generated graded A-modules induces a derived functor R= : D b Ž f ? g ? gr ? A-mod. ª D b Ž f ? g ? gr ? A !-mod. ,

Ž 26 .

which is a natural equivalence between the bounded derived categories of finitely generated graded modules over A and over A!. This is the contravariant version of the Koszul duality results from w5, 17, 22x which are generalized in w2, 3x. To make an explicit comparison, if the bigraded components =Ž C . ji are finite dimensional then the graded dual of the chain ˘., complex =ŽC. is naturally isomorphic to the cochain complex ‘G’ŽC j i ˘ ˘ where C is the cochain complex obtained from C by setting Ci s Cyj and ‘G’ is one of the covariant Koszul duality functors in the notation of w3, 5x. Let R i= denote the functor from D b Žf ? g ? gr ? A-mod. to the category of graded A!-modules defined on the chain level by taking R i=ŽC. to be ith homology Hi Ž =ŽC.. of the chain complex =ŽC.. Given any finitely generated graded A-module M, we can think of M as a chain complex C s Ž0 ª M ª 0. with C0 s M and trivial boundary. In view of Ž25., the graded A!-module R i=Ž M . s Ý j R i=Ž M . j can be described componentwise by j

R i= Ž M . s Ext Ajyi Ž M, K . yj

Ž 27 .

in terms of the standard grading on Ext A? Ž M, K .. If a graded A-module M s Ý iG q Mi with leading term in degree q has a finite free linear presentation A mK Y ª A mK Mq ª M ª 0 as in Ž2., then it is clear from the form of the dual presentation A!mK Y Hª A!mK MqU ª M = ª 0 that the linear Koszul dual M = is the graded A!-module R q=Ž M .. In particular, if we take M to be the ideal I p of p = p-minors then I p= is R p=Ž M .. From the natural isomorphisms U

ToriA Ž K , I p . ( Ext Ai Ž I p , K . it follows from the way Lascoux’s resolution is built from the complexes Wkp in Ž3. of Section 1 that in the case p ) 1 R i= Ž I p . s

½

X kp 0

U

if i s kp y k q 1 otherwise,

for 1 F k F l

Ž 28 .

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where l s minŽ m, n. y p q 1 and X kp s K mA Wkp. Consequently, every nonzero R i=Ž I p . for p ) 1 will be an atypical simple g-module. The interpretation of these modules as linear Koszul duals of primary GLŽ F . = GLŽ G .-invariant ideals Il is discussed in Section 5. 3. SCHUR SUPERMODULES AND THE DOUBLE COMPLEX CONSTRUCTION The goal of this section is to describe X kp as the homology of a double complex built from Schur supermodules. We begin with a brief preliminary discussion of ordinary Schur modules, Young symmetrizers, and related combinatorics Žsee w1, 4, 8, 15, 16, 18, 21x.. A sequence l s Ž l1 , . . . , l r . of nonnegative integers is said to be a partition of k, written l & k, if

l1 G l2 G ??? G l r and

l1 q l2 q ??? ql r s k ;

Ž 29 .

the sum l1 q l2 q ??? ql r of the ‘‘parts’’ of l is called the weight of l and is denoted by < l <. As is customary, we will consider nonincreasing sequences Ž l1 , . . . , l r , 0. and Ž l1 , . . . , l r . as representing the same partition. The Young ‘‘diagram’’ or ‘‘frame’’ associated to a partition l may be defined as a subset Dl s  Ž i , j . g N = N ¬ 1 F j F l i 4

Ž 30 .

of N = N and the ‘‘transpose’’ or ‘‘conjugate’’ partition of l as the ˜ s Ž l˜1 , l˜2 , . . . . whose Young diagram Dl˜ is the transpose partition l Ž j, i . ¬ Ž i, j . g Dl4 of the digram Dl . Given a partition l of k, let T be a Young tableau 4 of shape l with distinct entries in  1, . . . , k 4 , i.e., a one-to-one function T : Dl ª  1, . . . , k 4 . Define two subgroups RŽT . and C ŽT . of the symmetric group Ý k of permutations of  1, . . . , k 4 as follows: R Ž T . s  s g Ý k ¬ ; Ž i , j . g Dl , s Ž T Ž i , j . . s T Ž i , jX . for some jX 4 . C Ž T . s  s g Ý k ¬ ; Ž i , j . g Dl , s Ž T Ž i , j . . s T Ž iX , j . for some iX 4 .

Ž 31 . Using these subgroups we define two Young symmetrizers, s

cT Ž l . s Ýs g R ŽT . Ýt g C ŽT . Ž y1 . ts t

dT Ž l . s Ýs g R ŽT . Ýt g C ŽT . Ž y1 . ts , 4

Young tableaux are also referred to as Young diagrams in some of the literature.

Ž 32 .

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in the group algebra QŽÝ k . over the rationals, where Žy1. s denotes the sign of a permutation s . These symmetrizers are absolutely primitive Žpseudo.-idempotents in QŽÝ k . and different choices of tableaux T of the same shape l give rise to equivalent idempotents cT Ž l. and dT Ž l., respectively. For the sake of definiteness,we let Tl : Dl ª  1, . . . , k 4 denote the tableau defined by TlŽ i, j . s l1 q ??? ql iy1 q j and we set cŽ l. s cTlŽ l., dŽ l. s dTlŽ l.. ŽThe symmetrizer cŽ l. is denoted eŽ l. in w21x.. Now let E be a vector space over a field K containing Q, and let Emk denote the tensor product E mK ??? mK E of k-copies of E over K. Viewing the Yound idempotents cŽ l., dŽ l. as symmetrizing operators on Emk through the action of the symmetric group Ý k on Emk by permutation of tensor factors, their range subspaces cŽ l. Emk , dŽ l. Emk are GLŽ E .-submodules of Emk . We define the Schur module LlŽ E . to be cŽ l. Emk and the Weyl module SlŽ E . to be dŽ l. Emk . Ž SlŽ E . is also called a co-Schur module and is denoted KlŽ E . in w1, 21x.. If E is a vector space of dimension r then LlŽ E . is an irreducible GLŽ E .-module, SlŽ E . is isomorphic to Ll˜Ž E ., and the set  SlŽ E . ¬ l s Ž l1 , . . . , l r . & k 4 is a complete set of nonisomorphic irreducible homogeneous polynomial representations of GLŽ E . of degree k. It is useful to have available the generalization of the above constructions to ‘‘skew’’ Schur and Weyl modules. Let : denote the partial order defined on the set of all partitions as follows:

l ; m m li F mi

for all i s 1, 2, . . . .

Ž 33 .

A ‘‘skew’’ partition lrm can be thought of as an ordered pair Ž l, m . of partitions such that m : l, its weight < lrm < is defined to be < l < y < m <, and the skew diagram Dl r m of shape lrm is defined to be Dl ]Dm . Given a skew partition lrm weight < lrm < s k, one can define skew Young symmetrizers cT Ž lrm ., dT Ž lrm . associated to a tableau T : Dl r m ª  1, . . . , k 4 of skew shape lrm with distinct entries in the same manner as done above for a single partition. One may then define GLŽ E .-modules Ll r mŽ E . s cŽ lrm . Emk , Sl r mŽ E . s dŽ lrm . Emk for a vector space E, where cŽ lrm ., dŽ lrm . are set to be the symmetrizers associated to the tableau Tl r m : Dl r m ª  1, . . . , k 4 defined by Tl r m Ž i , j . s l1 q ??? ql iy1 q j y Ž m 1 q ??? qm iy1 q m i . .

Ž 34 .

These GLŽ E .-modules satisfy the relation Sl r mŽ E . ( Ll˜ r m˜Ž E . but they are not irreducible in general. ŽSee w21, 0.3Žc.x for another description of Ll r mŽ E . and for the decomposition of Ll r mŽ E . into irreducibles by the

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‘‘adjoint’’ version of the Littlewood]Richardson formula.. It is often convenient to think of SlŽ E ., LlŽ E . as special cases of Sl r mŽ E ., Ll r mŽ E ., where m is taken to be the zero sequence Ž0, . . . , 0., the so-called empty partition f . As in Section 2, let V s V0 [ V1 be a Z 2-graded vector space over a field K of characteristic zero, where V0 s F and V1 s G are of dimension m and n, respectively. The action of the symmetric group Ý k on the tensor products ¨ 1 m ??? m ¨ k g Vr 1 m ??? m Vr k of homogeneous vectors by r

s Ž ¨ 1 m ??? m ¨ k . s Ž y1 . ¨sy1 Ž1. m ??? m ¨sy 1 Ž k . ,

Ž 35 .

where r s Ý i- j, s Ž i.) s Ž j. ri r j , which clearly commutes with the glŽ m ¬ n.action 5 defined by k

f Ž ¨ 1 m ??? m ¨ k . s

Ý Ž y1. s r q ??? qr Ž

1

iy 1 .

¨ 1 m ???

is1

m ¨ iy1 m f Ž ¨ i . m ¨ iq1 m ??? m ¨ k

Ž 36 .

for a homogeneous element f g glŽ m ¬ n. of degree s. One can then repeat the constructions with Young symmetrizers in this setting to define glŽ m ¬ n.-modules Ll r mŽ V . and Sl r mŽ V . as the submodules cŽ lrm .V mk and dŽ lrm .V mk of V mk , respectively. As before, we have isomorphisms Sl r mŽ V . ( Ll˜ r m˜Ž V . as glŽ m ¬ n.-modules. Moreover, since the image of the enveloping algebra UŽglŽ m ¬ n.. in the endomorphism algebra Hom K Ž V mk , V mk . is the centralizer of the action of the group Ý k on V mk , the absolute primitivity of the symmetrizers cŽ l., dŽ l. implies that the glŽ m ¬ n.-modules LlŽ V ., SlŽ V . are irreducible Žsee w4x.. If we restrict the action of glŽ m ¬ n. on the Schur supermodule Ll r mŽ V . to the even subalgebra gl 0 Ž m ¬ n. s glŽ F . = glŽ G . then we have the direct sum decompositions Ll r m Ž V . (

Ý Ll rg F m Sg r m G ( Ý Sl rg G m Lg r m F , g

Ž 37 .

g

where the summations run over all partition g satisfying m : g : l. These decompositions can be viewed as special cases of the decomposition Že. in Section 0.3 of w21x for Schur complexes. It may be worthwhile to observe that Schur complexes over a commutative ring R containing a field K of characteristic zero can be interpreted in terms of Schur supermodules as follows: given a complex F of free R-modules and a skew partition lrm of 5

See the beginning of Section 4 for a general discussion of tensor products of Lie supermodules.

IDEALS AND REPRESENTATIONS OF

glŽ m ¬ n.

571

weight < lrm < s k, the Schur complex Ll r mŽF. is the subspace cŽ lrm .Fmk of Fmk , where Fmk is viewed as a Z 2-graded vector space, which can easily be seen to a R-subcomplex of the tensor product complex Fmk . ŽSee Section 0.3 of w21x about Schur complexes.. Before we describe the double complex of glŽ m ¬ n.-modules whose homology is the reduction X kp s K mS Wkp of the minimal free complex Wkp appearing in Lascoux’s resolution of I p over the symmetric algebraic algebra S s SK Ž F U m G ., we need to recall from w21x the double complex L?k,? p of free chain complexes over S whose homology is the complex Wkp. Taking FS s S mK F and GS s S mK G to be free S-modules of ranks m and n, respectively, we let w : GS ª FS denote the generic map free S-modules, described in terms of bases  f 1 , . . . , f m 4 for F and  g 1 , . . . , g n4 for G by n

w Ž gi . s

Ý f jU m g i m f j g S1Ž F U m G . m F ,

Ž 38 .

js1

where  f iU , . . . , f mU 4 is the basis of F U dual to  f 1 , . . . , f m 4 . As in w21x, we let Ž t . r denote the rectangular partition Ž t, t, . . . , t . where the entry t repeats r times. We let L?k,? p denote the double complex L? ?ŽŽ m y p y k q 1. k , Ž n y p y k q 1. k ; w . constructed in Definition 1.6 of w21x and denoted there as L?k ? with p suppressed in the superscript. L?k,? p is a double complex 0 0 x x ??? ª L1,k ,0p ª L 0,k ,0p ª0 x x k, p ??? ª L1, 1 ª L 0,k ,1p ª0 x .. .

Ž 39 .

x .. .

of free chain complexes Ls,k,t p over S and these can be described in terms of Schur complexes as follows: Ls,k t, p s [

Ý Ý Ln _ l r m Ž w . m Ln _ g r m Ž w U . .

Ž 40 .

m&s n &t

The morphism w : GS ª FS and its dual w U : FSU ª GUS are being considered as chain complexes that are nontrivial in degrees 1 and 0, l and g denote the rectangular partitions Ž m y p y k q 1. k and Ž m y p y k q 1. k , respectively, n _ lrm and n _ grm denote the skew partitions Ž l1 y n k , . . . , l k y n 1 .rm and Žg 1 y n k , . . . , g k y n 1 .rm, respectively, and the double sum is over all partitions m & s and n & t for which these skew

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partitions make sense. ŽFor a pictorial version of Ž40., see Proposition 1.9 of w21x, where the notation lrn = m is used for n _ lrm.. The differentials of L?k,? p are described explicitly in the proof of Theorem 1.8 of w21x. We write the total complex of L?k,? p as a chain complex k, p L?k , p s Ž ??? ª Liq1 ª Li k , p ª ??? . ,

Ž 41 .

where Li p, k denotes the direct sum Ý sytsi Ls,k,t p. DEFINITION 3.1. We let L?k,? p denote the double complex K mS L?k,? p of graded vector spaces obtained by reducing L?k,? p modulo the ideal Sqs I1. The total complex of L?k,? p is the chain complex L?k, p of graded vector spaces K m L?k, p. PROPOSITION 3.2. Ža. The homology of the total complex L?k, p of the double complex L?k,? p is concentrated in degree 0, and in fact in bidegree Ž0, 0.. Žb. H0 Ž L?k, p . is naturally isomorphic to X kp. Proof. Corollary 2.13 of w21x states that the homology of the total complex L?k, p of the double complex L?k,? p is concentrated in degree 0 and is in fact a subfactor of the term L 0,k,0p of bidegree Ž0, 0.. By Theorem 2.14 of w21x, H0 Ž L?k, p . is free over S, so that H Ž K mS L?k, p . ( K mS H Ž L?k, p . and statement Ža. follows. Then, keeping in mind w21, 2.17x and w21, 4.9x, we have that H0 Ž L?k, p . s H0 Ž K m L?k, p . ( K m H0 Ž L?k, p . s K m Wkp s X kp. COROLLARY 3.3. to the direct sum

Each bigraded component Ls,k,t p is naturally isomorphic U Ý Ý Ln _ l r m Ž V . m Ln _ g r m Ž V w1x .

Ž 42 .

m&s n &t

in the notation of Ž40., where V w1xU denotes the linear dual of the superspace V w1x obtained by shifting the basic superspace V. Under the natural glŽ m ¬ n.module 6 structure acquired by each component Ls,k,t p through these isomorphisms, the double complex L?k,? p of graded ¨ ector spaces becomes a double complex of glŽ m ¬ n.-modules. Proof. The decomposition is just the reduction of Ž41. modulo K. From the description of the differentials of the double complex L?k,? p given in the proof of w21, 1.8x, in order to prove that the differentials of L?k,? p are glŽ m ¬ n.-maps, it is sufficient to check the glŽ m ¬ n.-equivariance of the 6 See the beginning of Section 4 for the description of the standard left-module structure M U s M U s on the linear dual of a Lie supermodule M.

IDEALS AND REPRESENTATIONS OF

glŽ m ¬ n.

573

trace and evaluation maps U

tr : K w 1 x ª V mK V w 1 x ,

U

ev : V m V w 1 x ª K w 1 x ,

Ž 43 .

which are reductions of tr : S ª w m w U , ev : w m w U ª S described in the introduction of w21x. The maps in Ž43. can be written in terms of bases  f 1 , . . . , f m 4 ,  g 1 , . . . , g n4 for F, G and their dual bases  f 1U , . . . , f mU 4 ,  g U1 , . . . , g U1 4 as follows: m

tr Ž 1 . s

Ý is1

ev Ž f i m

f jU

n

f i m f iU y

. s di j , ev Ž g i m

Ý g j m g Uj js1

g Uj

. s di j , ev ' 0

on F m GU [ G m F U .

Ž 44 . The verification that they are glŽ m ¬ n.-maps is simply a matter of keeping track of signs in the recipe for the glŽ m ¬ n.-action on the superspace V m V w1xU . COROLLARY 3.4. The chain complex L?k, p is a complex of glŽ m ¬ n.modules and consequently X kp acquires a glŽ m ¬ n.-module structure through the natural isomorphism X kp ( H0 Ž L?k, p .. Proof. L?k, p s K m L?k, p is the total complex of the double complex of glŽ m ¬ n.-modules.

L?k,? p

Remark 3.5. The glŽ m ¬ n.-module structure on X 1p arising from the presentation in Ž20. of Section 2, discussed further in Section 5, agrees with the glŽ m ¬ n.-module structure on X 1p in 3.4 up to multiplication by the supertrace character, i.e., up to tensoring by the one-dimensional supertrace module Lm F m Ln GU . More generally, the glŽ m ¬ n.-module structure on X kp arising from the point of view of linear Koszul duals agrees with the structure from 3.4 up to multiplication by the kth power of the supertrace character. Since the supertrace character is trivial over slŽ m ¬ n. there is no difference between the two structures as slŽ m ¬ n.modules, and we will restrict our attention to slŽ m ¬ n. whenever there is no reason to distinguish between the two module structures on X kp. 4. THE IRREDUCIBILITY OF X kp We begin this section by recalling some basic constructions with Lie supermodules, that is on Z 2-graded modules over a Lie superalgebra. Let g be a Lie superalgebra over a field K and let M, N be supermodules over g. The tensor product M mK N with the usual Z 2-grading becomes a

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AKIN AND WEYMAN

g-supermodule under the g-action defined by a Ž x m y . s ax m y q Ž y1 .

degŽ a . degŽ x .

x m ay

Ž 45 .

on homogeneous elements a g g, x g M, y g N. The g-supermodules M mK N and N mK M are isomorphic under the interchange map x m y ¬ Ž y1 .

degŽ x . degŽ y .

ymx

Ž 46 .

on homogeneous x g M, y g N. The dual space M U s Hom K Ž M, K . is right g-supermodule under the standard right action given by Ž x a.Ž x . s x Ž ax . for x g M U , a g g, x g M. Given any super anti-isomorphism u : g ª g of Lie superalgebras, we let M U u denote the left g-supermodule defined by the left g-action on M U given by a Ž x . s Ž y1 .

degŽ x . degŽ a .

Ž x . Ž u Ž a. .

Ž 47 .

on homogeneous a g g, x g M U . In the case of the standard Lie super anti-isomorphism s : g ª g defined by s Ž a. s ya for all a g g, it is customary to let M U stand for the left g-module M U s when no confusion is possible. In this paper we will also make use of another dual construction using the supertranspose. For g s glŽ m ¬ n. or slŽ m ¬ n., let t : g ª g be the supertranspose anti-isomorphism which can be defined in terms of block matrices by

t

ž

A C

B D

/

s

At yB t

Ct Dt

Ž 48 .

where Ž A, D . g g 0 , B g g 1 , and C g gy1. The dual module M U t has the same weights as M, with respect to the diagonal subalgebra of g, unlike M U s M U s which has negatives of the weights of M. PROPOSITION 4.1. The double complex L?k,? p is self-dual in the sense that the dual complex Ž L?k,? p .U t is isomorphic to L?k,? p as a double complex of glŽ m ¬ n.-modules. Proof. For any finite dimensional Z 2-graded glŽ m ¬ n.-module M, the modules Ll r mŽ M . and Sl r m˜ Ž M . are isomorphic, because cl r m and dl˜ r m˜ are equivalent idempotents in the group algebra of the symmetric group. Also, it follows easily from definitions that Ll r mŽ M .U t is isomorphic to Sl˜ r m˜Ž M U t . and hence to Ll r mŽ M U t . for any finite dimensional M. In particular, if we take M to be the basic glŽ m ¬ n.-module V or its shifted dual V w1xU s Ž V w1x.U s then we have t

Ll r m Ž V .U ( Ll r m Ž V . ,

U Ut

Ll r m Ž V w 1 x

U

. ( Ll r m Ž V w 1 x .

Ž 49 .

IDEALS AND REPRESENTATIONS OF

glŽ m ¬ n.

575

because M U t ( M in each of the cases M s V or M s V w1xU . Combining Ž49. with the decomposition Ž42. in Corollary 3.3, we find that Ž Ls,k,t p .U t and Ls,k,t p are isomorphic as glŽ m ¬ n.-modules. The proposition now follows from the fact that the trace and evaluation maps in Ž43. are duals of each other with respect to the duality functor Žy.U t , bearing in mind the sign rules in the interchanges of tensor factors for superspaces. As a consequence we obtain the following. COROLLARY 4.2. The glŽ m ¬ n.-module X kp is self-dual with respect to Žy.U t in the sense that Ž X kp .U t ( X kp as supermodules o¨ er glŽ m ¬ n.. It U follows that the dual module X kp s Ž X kp .U s is also self-dual with respect to Žy.U t . Let g denote either glŽ m ¬ n. or slŽ m ¬ n., h the diagonal subalgebra of g, b the Borel subalgebra consisting of the upper triangular matrices in g, and n the nilpotent subalgebra of strictly upper triangular matrices. With this setup, we say that a g-module is of highest weight if it is generated over g by a highest weight vector, i.e., an h-eigenvector that is annihilated by n. LEMMA 4.3. Let M be a finite dimensional g-module and suppose that M is a highest weight module of highest weight x . Then M is a simple g-module if and only if M U t is isomorphic to M as a g-module. Proof. Let Mx denote the x-eigenspace of M. By the assumption on M, the weight subspace Mx generates M, is annihilated by n, and is necessarily one-dimensional. If M is simple then the simple module M U t is isomorphic to M because isomorphism classes of finite dimensional simple g-modules are determined by their highest weights. Conversely, since MrradŽ M . is isomorphic to the simple g-module V Ž x . of highest weight x , if M U t ( M then the socle of M is isomorphic to V Ž x .U t ( V Ž x .. Therefore the socle of M contains the one dimensional weight subspace Mx and hence is all of M, as Mx generates M. U

THEOREM 4.4. The glŽ m ¬ n. modules X kp and X kp are irreducible. Proof. The glŽ m ¬ n.-module structure we put on X kp s K m Wkp must be tensored by k copies of the one-dimensional superdeterminant representation n m F m n n GU of glŽ m ¬ n. in order to be compatible with the structure coming from homological considerations. Therefore it will be more convenient to work over g s slŽ m ¬ n. throughout this proof. Since Wkp is a linear chain complex of graded free S-modules and X kp s K mS Wkp , we can think of Wkp as S mK X kp with the differential Wk,p iq1 ª Wk,p i of

576

AKIN AND WEYMAN

S-degree one being the composite

Ý S j m S1 m X kp, i

j

mm1

Ý S jq1 m X k , i , Ž 50.

6

1m ­ kp, iq1Ž0 .

6

Ý S j m X kp, iq1

j

j

where m denotes the restriction of the multiplication map S mK S ª S and ­ k,k iq1Ž0. : X k,p iq1 ª S1 m X k,p i is the restriction of the differential ­ k,p iq1 : Wk,p iq1 ª Wk,p i to the leading component S0 m X k,p iq1 of the graded module Wk,p iq1 s S m X k,p iq1. We claim that the complex Wkp is linearly injecti¨ e in the sense that the components ­ k,p iq1Ž0. : X k,p iq1 ª S1 m X k,p i of the differentials are injective for all i G 0. This claim can be proved by a comparison of the Roberts complex with the Lascoux resolution as follows.7 Using notation compatible with our description of Lascoux’s resolution, we can write the minimal chain complex constructed by Roberts in w23x as the ˘ ?p of a graded GLŽ F . = GLŽ G.-equivariant complex total complex W p ˘ 1p ª W ˘ ly1 ˘ 1p ª S ª 0 0ªW ª ??? ª W

Ž 51 .

˘k,p i ( S mK X k,p i , of chain complexes of graded free S-modules such that W with all differentials having the same degrees as their analogues in the ˘ kp in the Roberts construction are Lascoux resolution. The complexes W p ˘ kp ª W ˘ ky1 linearly injective and the differentials d˘kp : W satisfy the further condition that the components p d kp, 0 Ž 0 . : X kp, 0 ª S p m X ky1, 2 ky2

Ž 52 .

are injective for all k G 1. Using the fact that the Lascoux complex ˘ kp and resolves SrI p , one can show by induction on k that the complexes W p Wk are isomorphic as follows. Considering the Roberts and Lascoux ˘ kp and complexes as total complexes of double complexes whose rows are W p p p ˘ ? ª W? on the total complexes Wk , respectively, the comparison map W ˘ kp ª Wkp at preserves the row filtrations and hence induces chain maps W 0 the E -level of the associated spectral sequence. For each k G 1, there is a commutative diagram 6

X kp, 0

p

p

d˘k , o Ž0 .

d k, 0 Ž0 .

Ž 53 .

6

p S p m X ky1, 2 ky2

6

6

p S p m X ky1, 2 ky2

X kp, 0

7 Another proof of this is given in w25x as a part of the proof of Theorem 5.2 stated in Section 5 of this paper.

IDEALS AND REPRESENTATIONS OF

glŽ m ¬ n.

577

which is a graded component of the diagram

˘ kp . ª H0 Ž Wkp . H0 Ž W x x p p ˘ H2 k Ž Wky1 . ªH2 k Ž Wky1 .

Ž 54 .

appearing at the E 1-level of the map on spectral sequences induced by the comparison map. Keeping in mind that X k,p 0 is the simple g 0-module LŽ pqky1. k F U m LŽ pqky1. kG ( LŽ mypykq1. k F m LŽ mypykq1. kGU , Ž 55 . it follows from the description given in Section 3 of w21x for the chain map p differential d kp : Wkp ª Wky1 that the vertical differential component p d kp, 0 Ž 0 . : X kp, 0 ª S p m X ky1, 2k

Ž 56 .

in the Lascoux double complex is nonzero and hence is injective for all k G 1, as has already been observed for the analogous component d˘k,p 0 Ž0. in Ž52. of the Roberts double complex. Since both vertical maps in Ž53. are injective and the bottom horizontal map is an isomorphism by induction on k, it follows that the top horizontal map in Ž53. is an isomorphism. ˘k,p 0 ª Wk,p 0 of the comparison map Consequently, the leading component W is an isomorphism. Next, for each j G 1 we consider the commutative diagram p

­˘k , j Ž0 .

S1 m X kp, jy1

6

X kp, j

(

6

6

p

Ž 57 .

S1 m X kp, jy1

6

X kp, j

­ km j Ž0 .

which is a graded component of the diagram p

­˘k , j Ž0 .

S1 m X kp, jy1

6

X kp, j

(

Ž 57 .

6

S1 m X kp, jy1

6

6 X kp, j

p ­ k , j Ž0 .

which is a graded component of the diagram p

˘kp, jy1 W (

6

p ­ k, j

Wkp, jy1

6

6 Wkp, j

­˘k, j

6

˘kp, j W

Ž 58 .

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AKIN AND WEYMAN

˘ kp ª Wkp. The injectivity of the top map in arising from the chain map W Ž57., by linear injectivity of the Roberts complex, and the bijectivity of the right vertical map in Ž57., by induction on j, imply that the left vertical map in Ž57. and hence the one in Ž58. is an isomorphism. It follows that ˘ kp and therefore that Wkp is linearly injective as Wkp is isomorphic to W claimed. In order to prove the theorem, it is sufficient in view of Ž4.2. and Ž4.3. to prove that X kpU is a highest weight module. We will do this by displaying a surjection from a Kac module V Ž x . onto X kpU . For convenience, we let M s Ý iG 0 Mi denote the g-module X kp with the Z-grading inherited from the complex Wkp , so that the leading term M0 is the simple g 0-module LŽ pqky1. k F U m LŽ pqky1. kG of highest weight yx where x g hU is the weight corresponding to the sequence ..., _ k , 0, . . . , 0,^yk,` . . . , yk ž ^k ,` _/ g Z pqky1

mq n

,

Ž 59 .

pqky1

which is the highest weight of M0U s LŽ pqky1. k F m LŽ pqky1. kGU . Under the action of g on M U s Ý i G 0 MiU s Ý iG 0 Ž M U .yi , gq1 obviously annihilates the term M0U so that the universal property of Kac modules implies the existence of a g-map D from the Kac module V Ž x . s Ind gp Ž M0U . into M U induced by the injection M0U ª M U as modules over the subalgebra p s g 0 [ g 1 of g. We want to show that the map D : V Ž x . ª M U is surjective, or equivalently that the dual map D* : M ª V Ž x .U is injective. Since V Ž x . is isomorphic as a p-module to HŽ gy1 . m M0 , we can think of U . mK M0 of Z-graded pyDU : M ª V Ž x .U as a map M ª HŽ gy1 U . mK M0 being modules, with the leading component D 0 : M0 ª H0 Ž gy1 an isomorphism. Consider the diagrams 1m DU iq1

6

trm1

trm1 1m DU iq1

U U gy1 mK gy1 mK Hiq1 Ž gy1 . mK M0 Ž 60 .

6

1m h

6

6

U gy1 mK Mi

1m h 1m DU i

6

6

U gy1 mK gy1 mK Miq1

U K mK Hiq1 Ž gy1 . m M0

6

K mK Miq1

U U gy1 m Hi Ž gy1 . mK M0

U where tr : K ª gy1 m gy1 is the formal trace map sending 1 to Ý s, t f sU m U g t m f t m g s in terms of basis elements as in the proof of Ž3.3., and h denotes the action of gy1 on the appropriate components of the pU . m M0 . The commutativity of the upper rectangle modules M and HŽ gy1

IDEALS AND REPRESENTATIONS OF

glŽ m ¬ n.

579

is trivial while that of the lower rectangle is a consequence of DU being a g-map. The composite of the rectangles in Ž60. is the commutative diagram 1m DU iq1

U Hiq1 Ž gy1 . mK M0

6

Miq1

6

1m DU i

U U gy1 mK Hi Ž gy1 . mK M0

6

6

U gy1 mK Mi

where the left vertical map is the leading component ­ k,p iq1Ž0. : X k,p iq1 ª S1 mK X k,p i of the boundary map ­ k,p iq1 : Wk,p iq1 ª Wk,p i , under the natural U identification of gy1 with g 1 s F U mK G s S1 , and the right vertical map U . mK M0 ª S1 mK Hi Ž g . mK M0 of is the leading component Hiq1 Ž gy1 iq1 Ž U the Koszul differential S mK H F mK G . ª S mK Hi Ž F U mK G . tensored by 1 M 0 . Since both vertical maps are injective, it follows by induction on i that DUiq1 is an injective as desired. 5. SOME EXAMPLES AND REMARKS In general, the larger the order p of minors is relative to the size of the m = n generic matrix Ž x i j ., the simpler is the structure of the Lascoux resolution of I p . The constructions in this paper can be illuminated by looking at cases of minors of relatively large order. In view of the symmetry between the roles played by the vector spaces F and G in slŽ F ¬ G . s slŽ m ¬ n. we will for simplicity assume that m G n, where m s dimŽ F . and n s dimŽ G .. In the case p s n of maximal order minors, the Lascoux resolution of I p consists of a single linear complex W1p p

m

0 ª R m L F U m Smy p G m L G ª ??? pq1

p

p

p

ª R m L FU m G L G ª R m L FU m L G of free R-modules over R s SŽ F U m G ., where the module of chains in homological degree i is the graded free R-module pqi

p

R m L F U m Si G m L G generated by the leading term L pq i F U m Si G m L p G in degree p q i over the graded ring R. This complex is the characteristic-zero version of the characteristic-free Eagon]Northcott resolution of the ideal I p s In of

580

AKIN AND WEYMAN

maximal order minors. ŽIn the characteristic-free version, the symmetric power Si G is replaced by the module Di G of symmetric tensors of degree i w1x.. Consequently, X 1p is the graded module whose homogeneous component in degree p q i is the irreducible GLŽ F . = GLŽ G .-module L pq i F U m S i G m L p G, the factor L p G s Ln G being one-dimensional. In order to see the slŽ F ¬ G .-module structure put on X 1p in Section 3, we recall in general that X 1p is the homology of the total complex of the double complex L?1,? p of slŽ F ¬ G .-modules and that the homology is concentrated in homological degree o. In the case p s n, Lo,1,op is the only nonzero term of the double complex L?1,? p and myp

nyp

L?1,? p s L Ž V . m L

myn

U

o

U

Ž V w1x . s L Ž V . m L Ž V w1x . ,

where V is the standard slŽ F ¬ G .-module F m G. Since Lo Ž V w1xU . is the trivial slŽ F ¬ G .-module K of dimension one, it follows that X 1n is just the exterior power Lmy n Ž V .. If we let p drop to n y 1, then the double complex L?1,? p has three nonzero terms ­ 1, 0

L 0,1,0p

6

L11,, 0p

­ 0, 1

6

L 0,1,1p so that X 1p is the middle and only homology of the complex 0 ª L1,1,0p ª L 0,1,0p ª L 0,1,1p ª 0, where L1,1,0p s Lmyn Ž V . s L 0,1,1p and L 0,1,0p s Lmynq1 Ž V . m V w1xU , or equivalently, X 1p is the homology of the sequence myn

­ 1, 0 mynq1

L ŽV . ¬

L

U

­ 0, 0 myn

Ž V . m V w1x ¸ L Ž V . .

Although the double complex L?1,? p gains more nonzero terms as we let p decrease below n y 1, Proposition 3.2 tells us that X 1p is always a subquotient of the term myp

nyp

L 0,1,0p s L Ž V . m L

U

Ž V w1x . .

More precisely, Proposition 3.2 says that the homology of L?1,? p is concentrated in bidegree Ž0, 0. which is equivalent to saying that X 1p is the quotient of kerŽ ­ 0, 0 . by imŽ ­ 1, 0 . l kerŽ ­ 0, 0 .. Since Proposition 3.2 covers L?k,? p for general k, the above observations extend to X kp for k ) 1, with

IDEALS AND REPRESENTATIONS OF

glŽ m ¬ n.

581

the reformulation that X kp is a subquotient of the slŽ F ¬ G .-module U

L 0,k ,0p s Ll Ž V . m Lg Ž V w 1 x

.,

where l and g are the rectangular partitions Ž m y p y k q 1. k and Ž m y p y k q 1. k , respectively. Our approach to putting an slŽ m ¬ n.-module structure on X 1p , or more generally X kp , has been through the double complex construction L?k,? p describing the components Wkp of the Lascoux resolution. Another point of view arises from looking at say X 1p for p ) 1 as the contravariant linear Koszul dual of the ideal I p , or equivalently, at X 1pU as the contravariant linear Koszul dual I p= of I p as described in Section 2. From this point of view, X 1pU s I p= for p ) 1 has a natural module structure over the nonpositive parabolic subalgebra pys gy1 [ g 0 of g s glŽ m ¬ n. of g s slŽ m ¬ n.; in fact, the presentation in Ž20. of Section 2 is a presentation of I p= by free py-modules. In w25x, we show that this is actually a presentation over g by direct sums of Kac modules, which in particular gives a direct way to extend the natural py-module structure on X 1pU to all of g. In order to state the more general version of this result proved in w25x, we first introduce some notation and terminology. Given a pair l s Ž l1 , . . . , l m . g Z m and m s Ž m 1 , . . . , m n . g Z n of partitions, the GLŽ F . = GLŽ G .-module SlŽ F . m SmŽ GU . is the simple modules of highest weight Ž l1 , . . . , l m , ym n , . . . , ym 1 . g Z mq n, where F, G are vector spaces of dimensions m, n over a field K of characteristic zero. For convenience we let Ž l ¬ m . denote the weight

Ž l1 , . . . , l m , ym n , . . . , ym 1 . and write V Ž l ¬ m . for the Kac module induced up to g s glŽ m ¬ n. from the simple module V0 Ž l ¬ m . s SlŽ F . mK SmŽ GU . over p s g 0 [ g 1. One of the equivalent conditions for atypicality given in w13x takes on the following form in this notation: the simple quotient V Ž l ¬ m . of V Ž l ¬ m . is atypical if and only if some term l i of l is conjugate to some term m j of m in the sense that l i y i s m j y j. The following is a special case of a more general result proved in w25x. THEOREM 5.1. Gi¨ en a pair l g Z m , m g Z n of partitions for which some l i is conjugate to some m j satisfying the further conditions that l i ) l iq1 and m j ) m jq1 , there exists a nonzero glŽ m ¬ n.-homomorphism from V Ž l ¬ m . into V Ž lX ¬ mX ., where

lX s Ž l1 , . . . , l iy1 , l i y 1, l iq1 , . . . , l m . X

and

m s Ž m 1 , . . . , m jy1 , m j y 1, m jq1 , . . . , m n . .

582

AKIN AND WEYMAN

In the special case where l g Z m and m g Z n represent the same partition g , i.e., lŽg 1 , . . . , gr , 0, . . . , 0. g Z m and m s Žg 1 , . . . , gr , 0, . . . , 0. g Z n, we shall write Vg 4 for V Ž l ¬ m . and Vg 4 for V Ž l ¬ m .. With the U notation, X kp ( Ž k . pqky14 and X kp ( VŽ p q k y 1. k 4 . The following result is proved in w25x. THEOREM 5.2. Let Ž s . t s Ž s, . . . , s . g Z t be a rectangular partition which is not a square, i.e,. s / t. Then there exists a linear free representation o¨ er HŽ gy1 . V

s, . . . , _ s , 1 . 5 m V ½ Ž s q 1,^` s, . . . , _ s . 5 ª V ½ Ž^` s, . . . , _ s . 5 ª V Ž s. 4 ª 0 ½ Ž^` t

t

ty1

t

which is a sequence of glŽ m ¬ n.-modules and VŽ s . t 4 is the linear Koszul dual IŽ=s. t of the primary g 0-in¨ ariant ideal IŽ s. t associated to the rectangular partition Ž s . t . U

This theorem says that if p ) 1 then X kp ( IŽ=k . pq ky 1, X kp ( IŽnpqky1. k U and that the linear presentation of X kp , X kp over HŽ gy1 . are actually glŽ m ¬ n.-presentations by Ždirect sums. of Kac modules. It is an important difficult problem to find generalizations of these presentations for other atypical simple g-modules, which may not always be linear and to try to extend the presentations to resolutions by modules which are, up to filtration, sums of Kac modules over g or at least to minimal free resolutions over UŽ gy1 .. Some partial results along these lines are given in w25x together with some general conjectures. In particular, the character formula for atypical simple glŽ m ¬ n.-modules conjectured in w10x is verified in w25x for the simple modules VŽ s . t 4 associated with rectangular partitions Ž s.t . REFERENCES 1. K. Akin, D. A. Buchsbaum, and J. Weyman, Schur functors and Schur complexes, Ad¨ . Math. 44 Ž1982., 207]278. 2. A. A. Beilinson, V. Ginsburg, and V. A. Schechtman, Koszul duality, J. Geom. Phys. 5 Ž1988., 317]350. 3. A. A. Beilinson, V. Ginsburg, and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 Ž1996., no. 2, 473]527. 4. A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Ad¨ . Math. 64 Ž1987., 118]175. 5. I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Algebraic bundles over P n and problems of linear algebra, Funct. Anal. Appl. 12 Ž1978., 66]67.

IDEALS AND REPRESENTATIONS OF

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6. I. N. Bernstein and D. A. Leites, A character formula for irreducible finite dimensional modules over the Lie superalgebras of series gl and sl, C. R. Acad. Bulgare. Sci. 33 Ž1980., 1059]1051. wIn Russianx 7. C. de Concini, D. Eisenbud, and C. Procesi, Young diagrams and determinantal varieties, In¨ ent. Math. 56 Ž1980., 129]165. 8. J. Dieudonne ´ and J. B. Carrell, Invariant theory, old and new, Ad¨ . Math. 4 Ž1971. wReprinted by Academic Press, New YorkrLondon, pp. 1]80.x 9. M. D. Gould, P. D. Jarvis, and A. J. Bracken, Branching rules for a class of typical and atypical representations of gl Ž m ¬ n., J. Math. Phys. 31 Ž1990., 2803]3810. 10. J. Van der Jeugt, J. W. B. Hughes, R. C. King, and J. Thierry-Mieg, Character formulas for irreducible modules of the Lie superalgebras sl Ž m ¬ n., J. Math. Phys. 31 Ž1990., 2278]2304. 11. V. G. Kac, Lie superalgebras, Ad¨ . Math. 26 Ž1977., 8]96. 12. V. G. Kac, Characters of typical representations of classical Lie superalgebras, Comm. Algebra 5 Ž1977., 889]897. 13. V. G. Kac, ‘‘Representations of Classical Lie Superalgebras,’’ ŽK. Bleuler, M. Petry, and A. Reetz, Eds.., Lecture Notes in Mathematics, Vol. 676, pp. 579]626, Springer-Verlag, Berlin, 1977. 14. R. C. King, ‘‘S-functions and Characters of Lie Algebras and Superalgebras,’’ ŽD. Stanton, Ed.., IMA Volumes in Mathematics and Its Applications, Vol. 19, pp. 226]261, SpringerVerlag, New York, 1990. 15. A. Lascoux, Polynomes symetriques, foncteurs de Schur, et Grassmanniennes, These, ´ ` Universite ´ Paris VII Ž1977.. 16. A. Lascoux, Syzygies de varietes ´ ´ determinantales, Ad¨ . Math. Ž1978., 202]237. 17. C. Lofwall, On the subalgebra generated by the one-dimensional elements in the Yoneda ¨ ext-algebra, in Lecture Notes in Mathematics Vol. 1183, pp. 291]338, Springer-Verlag, Berlin, 1986. 18. H. A. Nielsen, ‘‘Tensor Functors of Complexes’’ Aarhus University Preprint Series No. 15, Aarhus University, Denmark, 1978. 19. I. Penkov and V. Serganova, Character formulas for some classes of atypical gl Ž m q n e . and pŽ m. modules, Lett. Math. Phys. 16 Ž1988., 251]261. 20. I. Penkov and V. Serganova, Cohomology of GrP for classical Lie supergroups G and characters of some atypical G-modules, Ann. Inst. Fourier 4 Ž1989., 845]873. 21. P. Pragacz and J. Weyman, Complexes associated with trace and evaluation. Another approach to Lascoux’s resolution, Ad¨ . Math. 57 Ž1985., 163]207. 22. S. B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 Ž1970., 39]60. 23. P. Roberts, A minimal free complex associated to the minors of a matrix, University of Utah Preprint, University of Utah, 1978. 24. A. N. Serge’ev, The tensor algebra of the standard representation as a module over the Lie superalgebra gl Ž m ¬ n. and QŽ n., Mat. Sbornik 123 Ž1984., 422]430. wIn Russianx 25. K. Akin and J. Weyman, Resolutions, Koszul duality, and representations of gl Ž m ¬ n., in preparation.