Presenting affine q-Schur algebras

We obtain a presentation of certain affine q-Schur algebras in terms of generators and relations. The presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine gln. Our results extend and rely on the corresponding result for the q-Schur algebra of the symmetric group, which were proved by the first author and Giaquinto. Mathematics Subject Classification (2000) 17B37, 20F55 Introduction Let V ′ be a vector space of finite dimension n. On the tensor space (V ′)⊗r we have natural commuting actions of the general linear group GL(V ′) and the symmetric group Sr . Schur observed that the centralizer algebra of each action equals the image of the other action in End((V ′)⊗r), in characteristic zero, and Schur and Weyl used this observation to transfer information about the representations of Sr to information about the representations of GL(V ′). That this Schur–Weyl duality holds in arbitrary characteristic was first observed in [4], although a special case was already used in [2]. In recent years, there have appeared various applications of the Schur–Weyl duality viewpoint to modular representations. The Schur algebras S(n,r) first defined in [9] play a fundamental role in such interactions. Jimbo [13] and (independently) Dipper and James [6] observed that the tensor space (V ′)⊗r has a q-analogue in which the mutually centralizing actions of GL(V ′) and Sr become mutually centralizing actions of a quantized enveloping algebra U(gln) and of the Iwahori-Hecke algebra H (Sr) corresponding to Sr. In S. R. Doty Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60626 U.S.A. E-mail: [email protected] R. M. Green Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO 80309 U.S.A. E-mail: [email protected] 2 S. R. Doty, R. M. Green this context, the ordinary Schur algebra S(n,r) is replaced by the q-Schur algebra Sq(n,r). Dipper and James also showed that the q-Schur algebras determine the representations of finite general linear groups in non-defining characteristic. An affine version of Schur–Weyl duality was first described in [3]. A different version, in which the vector space V ′ is replaced by an infinite dimensional vector space V , is given in [11], and we follow the latter approach here. In the affine (type A) setting, the mutually commuting actions are of an affine quantized enveloping algebra U(ĝln) and an extended affine Hecke algebra H (Ŵ ) corresponding to an extended affine Weyl group Ŵ containing the affine Weyl group W of type Âr−1. The affine q-Schur algebra Ŝq(n,r) in this context, which is also infinite dimensional, was first studied in [11], [17], and [20]. Recently, a new approach to Schur algebras or their q-analogues was given in [7], where it was shown that they may be defined by generators and relations in a manner compatible with the usual defining presentation of the enveloping algebra or its corresponding quantized enveloping algebra. The purpose of this paper is to extend that result to the affine case — that is, to describe the affine q-Schur algebra Ŝq(n,r) by generators and relations compatible with the defining presentation of U(ĝln). This result is formulated in Theorem 1.6.1, under the assumption that n > r. An equivalent result, which describes the affine q-Schur algebra as a quotient of Lusztig’s modified form of the quantized enveloping algebra, is given in Theorem 2.6.1. These results depend on a different presentation, also valid for n > r, of the q-Schur algebra given in [11, Proposition 2.5.1]. A different approach to the results of this paper seems to be indicated for the case n ≤ r. The organization of the paper is as follows. In Section 1 we give necessary background information, and formulate our main result. In Section 2 we give the proof of Theorem 1.6.1, and we also give, in Section 2.6, the alternative presentation mentioned above. Finally, in Section 3 we outline the analogous results in the classical case, when the quantum parameter is specialized to 1. After we submitted this paper, McGerty informed us that he has independently proved Theorem 2.6.1 using different methods; see [19]. 1 Preliminaries and statement of main results Our main result, stated in §1.6, is a presentation by generators and relations of the affine q-Schur algebra. In order to put this result in context, we review some of the definitions of the algebra that have been given in the literature. 1.1 Affine Weyl groups of type A The affine Weyl group will play a key role, both in our definitions and our methods of proof, so we define it first. The Weyl group we consider in this paper is that of type Âr−1, where we intend r ≥ 3. This corresponds to the Dynkin diagram in Figure 1.1.1. The number of vertices in the graph in Figure 1.1.1 is r, as the top vertex (numbered r) is regarded as an extra relative to the remainder of the graph, which is a Coxeter graph of type Ar−1. Presenting affine q-Schur algebras 3 Fig. 1.1.1 Dynkin diagram of type Âr−1 We associate a Weyl group, W =W (Âr−1), to this Dynkin diagram in the usual way (as in [12, §2.1]). This associates to node i of the graph a generating involution si of W , where sis j = s jsi if i and j are not connected in the graph, and sis jsi = s jsis j if i and j are connected in the graph. For t ∈ Z, it is convenient to denote by t the congruence class of t modulo r, taking values in the set {1,2, . . . ,r}. For the purposes of this paper, it is helpful to think of the group W as follows, based on a result of Lusztig [15]. (Note that we write maps on the right when dealing with permutations.) Proposition 1.1.2 There exists a group isomorphism from W to the set of permutations of Z satisfying the following conditions: (i + r)w = (i)w+ r (a) r ∑ t=1 (t)w = r ∑ t=1 t (b) such that si is mapped to the permutation t 7→    t if t 6= i, i +1, t −1 if t = i +1, t +1 if t = i,