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Presenting Affine Q-Schur Algebras

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Presenting Affine Q-Schur Algebras Mathematische Zeitschrift manuscript No. (will be inserted by the editor) S. R. Doty R. M. Green · Presenting affine q-Schur algebras Received: date / Revised: date Abstract 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 r Let V ′ be a vector space of finite dimension n. On the tensor space (V ′)⊗ 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 r image of the other action in End((V ′)⊗ ), 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 dual- ity 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 applica- tions 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 ten- r sor space (V ′)⊗ 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(gln) and an extended affine Hecke algebra H (W) corresponding to an extended affine Weyl group W containing the affine Weyl group W of type Ar 1. The affinec q-Schur algebra Sq(n,r) in this context, whichb is also infinite dimensional,− was first studied in [11],b [17], and [20]. b Recently, a new approach to Schurb 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 Sq(n,r) by generators and relations compatible with the defining presentation of U(gln). This result is formulated in Theorem 1.6.1, under the assumption that n > rb. An equivalent result, which describes the affine q-Schur algebra as a quotient of Lusztig’sc 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 presenta- tion 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 Ar 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 isb 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 Ar 1 − b We associate a Weyl group, W = W(Ar 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 notb 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 permu- tations of Z satisfying the following conditions: (i + r)w =(i)w + r (a) r r ∑(t)w = ∑ t (b) t=1 t=1 such that si is mapped to the permutation t if t = i,i + 1, 6 t t 1 if t = i + 1, 7→ − t + 1 if t = i, for all t Z. ∈ For reasons relating to weight spaces which will become clear later, we con- sider a larger group W of permutations of Z. ρ Definition 1.1.3 Letb be the permutation of Z taking t to t + 1 for all t. Then the group W is defined to be the subgroup of permutations of Z generated by the group W and ρ. As willb become clear later, the point of ρ is that conjugation by ρ will corre- spond to a graph automorphism of the Dynkin diagram given by rotation by one place. Proposition 1.1.4 (i) There exists a group isomorphism from W to the set of per- mutations of Z satisfying the following conditions: (i + r)w =(i)w + r b (a) 4 S. R. Doty, R. M. Green r r ∑(t)w ∑ t mod r. (b) t=1 ≡ t=1 (ii) Any element of W is uniquely expressible in the form ρzwforz Z and w W. ∈ ∈ Conversely, any element of this form is an element of W. S (iii) Let S ∼= r be theb subgroup of W generated by b s1,s2,...,sr 1 . { b − } Let Z be the subgroup of W consisting of all permutations z satisfying (t)z t mod r b ≡ Zr for all t. Then ∼= Z, Z is normal in W, and W is the semidirect product of S and Z. Proof The three parts are proved in [11,b Propositionb 1.1.3, Corollary 1.1.4, Propo- sition 1.1.5] respectively. ⊓⊔ It is convenient to extend the usual notion of the length of an element of a Coxeter group to the group W in the following way. Definition 1.1.5 For w W the length ℓ(w) of w is the length of a word of minimal ∈ b length in the group generators si of W which is equal to w. The length, ℓ(w′),ofa typical element w = ρzw of W (where z Z and w W) is defined to be ℓ(w). ′ ∈ ∈ When the affine Weyl group is thought of in the above way, the familiar no- tions of length and distinguishedb coset representatives may be adapted from the corresponding notions for Coxeter groups. Definition 1.1.6 Let Π be the set of subsets of S = s1,s2,...,sr , excluding S { } itself. For each π Π, we define the subgroup Wπ of W to be that generated by ∈ si π . (Such a subgroup is called a parabolic subgroup.) We will sometimes { ∈ } write Wπ for Wπ to emphasize that it is a subgroupb ofbW. Let Π ′ be the set of elements of Π that omit the generator sr. b All the subgroups Wπ are subgroups of W, and are parabolic subgroups in the usual sense of Coxeter groups. Furthermore, each such Wπ is isomorphic to a direct product of Coxeter groupsb of type A (i.e., finite symmetric groups) corresponding to the connected components of the Dynkin diagram obtainedb after omitting the elements si that do not occur in π.
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