Presenting cyclotomic q-Schur algebras

We give a presentation of cyclotomic q-Schur algebras by generators and defining relations. As an application, we give an algorithm for computing decomposition numbers of cyclotomic q-Schur algebras. § 0. Introduction Let Hn,r be an Ariki-Koike algebra associated to a complex reflection group Sn ⋉ (Z/rZ). A cyclotomic q-Schur algebra Sn,r associated to Hn,r, introduced in [DJM], is defined as an endomorphism algebra of a certain Hn,r-module. In this paper, we give a presentation of cyclotomic q-Schur algebras by generators and defining relations. In the case where r = 1, Hn,1 is the Iwahori-Hecke algebra of the symmetric group Sn, and Sn,1 is the q-Schur algbera of type A. In this case, Sn,1 is realized as a quotient algebra of the quantum group Uq = Uq(glm) via the Schur-Weyl duality between Hn,1 and Uq in [J]. We remark that the Schur-Weyl duality holds not only over Q(q) but also over Z[q, q] (see [Du]). By using the surjection from Uq to Sn,1, Doty and Giaquinto gave a presentation of Sn,1 by generators and defining relations in [DG]. They also gave a presentation of Sn,1 in the way compatible with Lusztig’s modified form of Uq. After that, Doty realized in [Do] the generalized q-Schur algebra (in the sense of Donkin) as a quotient algebra of a quantum group (also Lusztig’s modified form) associated to any Cartan matrix of finite type. In the case where r > 1, a Schur-Weyl duality between Hn,r and Uq(g) over K = Q(q, γ1, · · · , γr) was obtained by Sakamoto and Shoji in [SakS], where g = glm1 ⊕ · · · ⊕ glmr is a Levi subalgebra of a parabolic subalgebra of glm. However, this Schur-Weyl duality does not hold over Z[q, q, γ1, · · · , γr]. In fact, SakamotoShoji’s Schur-Weyl duality should be understood as a Schur-Weyl duality between modified Aliki-Koike algebra H 0 n,r introduced in [S1], and Uq(g) rather than the duality between Hn,r and Uq(g). The image of Uq(g) in the Schur-Weyl duality is isomorphic to the modified cyclotomic q-Schur algebra S 0 n,r associated to H 0 n,r introduced in [SawS]. H 0 n,r and S 0 n,r are defined over any integral domain R with parameters satisfying certain conditions. In particular, we have Hn,r ∼= H 0 n,r over K though S 0 n,r 6 ∼= Sn,r. (Note that Hn,r 6∼= H 0 n,r over R in general.) Some relations between Sn,r and S 0 n,r were studied in [SawS] and [Saw]. They showed that S 0 n,r turns out to be a subquotient algebra of Sn,r, and S 0 n,r ∼= ⊕ (n1,··· ,nr) n1+···+nr=n Sn1,1 ⊗ ∗This research was supported by JSPS Research Fellowships for Young Scientists 1