CYCLOTOMIC q-SCHUR ALGEBRAS ASSOCIATED TO THE ARIKI-KOIKE ALGEBRA

Let Hn,r be the Ariki-Koike algebra associated to the complex reflection group Sn (Z/rZ)n, and let S(Λ) be the cyclotomic q-Schur algebra associated to Hn,r, introduced by Dipper, James and Mathas. For each p = (r1, . . . , rg) ∈ Zg>0 such that r1 + · · · + rg = r, we define a subalgebra Sp of S(Λ) and its quotient algebra S. It is shown that Sp is a standardly based algebra and S is a cellular algebra. By making use of these algebras, we prove a product formula for decomposition numbers of S(Λ), which asserts that certain decomposition numbers are expressed as a product of decomposition numbers for various cyclotomic q-Schur algebras associated to ArikiKoike algebras Hni,ri of smaller rank. This is a generalization of the result of N. Sawada. We also define a modified Ariki-Koike algebra H of type p, and prove the Schur-Weyl duality between H and S. 0. Introduction Let H = Hn,r be the Ariki-Koike algebra over an integral domain R associated to the complex reflection group Wn,r = Sn (Z/rZ) n with parameters q,Q1, . . . , Qr ∈ R such that q is a unit in R. Let P̃n,r (resp. Pn,r) be the set of r-compositions (resp. r-partitions) of n. The cyclotomic q-Schur algebra S(Λ) associated to H was introduced by Dipper, James and Mathas [DJM], which is the endomorphism algebra of a certain H-module M = ⊕ μ∈Λ M , where Λ is a saturated subset of P̃n,r. They showed that S(Λ) is a cellular algebra in the sense of Graham and Lehrer [GL], and Mathas [M] showed that the Schur-Weyl duality (i.e., the double centralizer property) holds between H and S(Λ) in the case where Λ = P̃n,r with a certain condition. On the other hand, the modified Ariki-Koike algebraH was introduced in [SawS], under the condition that (*) “Qi − Qj are units in R for each i = j”, based on the study of the Schur-Weyl duality between H and a certain subalgebra of the quantum group of type A ([SakS], [Sh]). By using the cellular structure of H, a cyclotomic q-Schur algebra associated to H was constructed, in analogy to S(Λ). It was shown in [SawS] that this cyclotomic q-Schur algebra is isomorphic to the quotient algebra S of a certain subalgebra S of S(Λ), and that the Schur-Weyl duality holds betweenH and S. Moreover, the structure theorem for Swas proved, which asserts that S is a direct sum of tensor products of various q-Schur algebras S(P̃ni,1) associated to the Iwahori-Hecke algebra of type Ani−1. Received by the editors November 1, 2007 and, in revised form, February 6, 2010. 2010 Mathematics Subject Classification. Primary 20C08, 20G43. c ©2010 American Mathematical Society Reverts to public domain 28 years from publication