The Kostant form of U(sl + n) and the Borel subalgebra of the Schur algebra S(n, r)

Let An(K) be the Kostant form of U(sl + n ) and Γ the monoid generated by the positive roots of sln. For each λ ∈ Λ(n, r) we construct a functor Fλ from the category of finitely generated Γ-graded An(K)-modules to the category of finite-dimensional S(n, r)-modules, with the property that Fλ maps (minimal) projective resolutions of the one-dimensional An(K)module KA to (minimal) projective resolutions of the simple S (n, r)module Kλ. Introduction The polynomial representations of the general linear group GLn(C) were studied by I. Schur in his doctoral dissertation [16]. In this famous work, Schur introduced the, now called, Schur algebras, which are a powerful tool to connect r-homogeneous polynomial representations of the symmetric group on r symbols. These results of I. Schur were generalised by J.-A. Green to infinite fields of arbitrary characteristic in [10]. In Green’s work the Schur algebra S(n, r) = SK(n, r) plays the central role in the study of polynomial representations of GLn(K). In [7] Donkin shows that S(n, r) is a quasi-hereditary and so it has finite global dimension. This led to the problem of describing explicit projective resolution of the Weyl modules for S(n, r). Only partial answers to this problem are known. In [1] and [19] such resolutions were constructed for the case when K is a field of characteristic zero. If K has arbitrary characteristic then projective resolutions of Wλ are given in [2] when n = 2 (λ arbitrary), and in [13] and [17] for hook partitions. In [17] Woodcock provides the tools to reduce the problem of constructing these resolutions to the similar problem for the simple modules for the Borel subalgebra S(n, r) of S(n, r). Financial support by CMUC/FCT gratefully acknowledged by both authors. The second author’s work is supported by the FCT Grant SFRH/BPD/31788/2006. 1 Denote by Λ(n, r) the set of compositions of r onto n parts. It is proved in [15] that all simple S(n, r)-modules are one-dimensional and parametrised by the set Λ(n, r). We denote the simple module corresponding to λ ∈ Λ(n, r) module by Kλ. In [15], the first two steps in a minimal projective resolution of Kλ and the first three terms of a minimal projective resolution in the case n = 2 are constructed. In [18] minimal projective resolutions for Kλ for λ ∈ Λ(2, r) and non-minimal projective resolutions of Kλ for λ ∈ Λ(3, r) are constructed. The results of both papers depend on heavy calculations in the algebra S(n, r). In the present paper we take a more abstract approach. Let us denote by An(K) the Kostant form over the field K of the universal enveloping algebra of the Lie algebra sl n of upper triangular nilpotent matrices. Then An(K) has a unique one-dimensional module, which we denote by KA. In this paper we show that the construction of (minimal) projective resolutions for Kλ is essentially equivalent to the construction of (minimal) projective resolution for KA. The last task is much more feasible, since an explicit presentation of An(K) can be given and thus the results of Anick [3] can be applied to the description of an explicit projective resolution of KA. It is also worth to note that An(K) is a projective limit of finite dimensional algebras in the case char(K) = p > 0, and therefore the technique developed in [5] can be used for the construction of the minimal projective resolution of KA. This line of research will be followed by us in the subsequent papers. The general plan of the present paper is as follows. In Section 1 we collect general technical results, which will be used in the following section. We believe that these results can be applied in more general context, in particular to the generalised and q-Schur algebras. Let G be an ordered group with neutral element ǫ. Denote by Γ ⊂ G the submonoid of non-negative elements of G. For every Γ-graded algebra A and every Γ-set S, we construct a family of Γ-algebras {C(X)|X ⊂ S} and a family of Γ-graded algebra homomorphisms { φX : C(Y ) → C(X) ∣ X ⊂ Y ⊂ S } such that φX ◦ φ Z Y = φ Z X for every triple X ⊂ Y ⊂ Z of subsets in S. In other words, C(−) is a presheaf of Γ-graded algebras. For every x ∈ S, we construct an exact functor Fx from the category C (A,Γ) of finitely generated Γ-graded A-modules to the category C (C(S),Γ). If Γ acts by automorphisms on S, then the functors Fx preserve projective modules, and thus map projective resolutions into projective resolutions. If Aǫ ∼= K, then the functors Fx map minimal projective resolutions into minimal projective resolutions. For every X ⊂ S, we can consider each left C(X)-module as a left C(S)module via a homomorphism φX : C(S) → C(X). Thus we get a natural inclusion of categories (φX) ∗ : C (C(X),Γ) → C (C(S),Γ). 2 There is a left adjoint functor to ( φX ∗ (φX)∗ = C(X)⊗C(S) − : C (C(S),Γ) → C (C(X),Γ), where we consider C(X) as a right C(S)-module via φX . The main objective of Section 1.4 is to get conditions on X ⊂ S, ensuring that for every left C(X)module M and every (minimal) projective resolution P• of ( φX ∗ (M) the complex ( φX ) ∗ (P•) is a (minimal) projective resolution of M ∼= ( φX ) ∗ ( φX ∗ (M). If the both algebras C(S) and C(X) are artinian and ( φX ) ∗ has the above mentioned property, then the ideal Ker ( φX ) is a strong idempotent ideal. The algebra C(X) is finite dimensional and thus artinian, if X is finite and A is locally finite dimensional. But the algebra C(S) is rarely finite dimensional. To cope with this, we take a two stage approach. We say that Y ⊂ S is Γ-convex, if from γ = γ1γ2 and x, γx ∈ Y it follows that γ2x ∈ Y . In Proposition 1.29, we show that if Y is a convex Γ-set then the functor ( φY ) ∗ is exact and maps (minimal) projective resolutions into (minimal) projective resolutions. Let Y be a finite Γ-convex subset of S and X a subset of Y . Suppose that A is locally finite dimensional. In Theorem 1.35 we give a criterion for Ker ( φX ) to be a strong idempotent ideal. In Section 2 we apply the results of Section 1 to S(n, r). In our particular case, the algebra A is the Kostant form An(K), the set S is Z n and Γ is the submonoid of Z generated by the elements (0, . . . , 1,−1, . . . , 0). Then we show in Theorem 2.20 that C(Λ(n, r)) ∼= S(n, r). Here we consider Λ(n, r) as a subset Z in the natural way. Note that this isomorphism gives a description of S(n, r), which is similar to the idempotent presentation of the algebra S(n, r) obtained by Doty and Giaquinto in [8]. The set of compositions Λ(n, r) is contained in the lager finite set Λ(n, r) defined by Λ(n, r) = { (z1, z2, . . . , zn) ∈ Z n ∣