On decomposition numbers of the cyclotomic q-Schur algebras

Let S(Λ) be the cyclotomic q-Schur algebra associated to the ArikiKoike algebra H . We construct a certain subalgebra S(Λ) of S(Λ), and show that it is a standardly based algebra in the sense of Du and Rui. S(Λ) has a natural quotient S(Λ), which turns out to be a cellular algebra. In the case where the modified Ariki-Koike algebra H ♭ is defined, S(Λ) coincides with the cyclotomic q-Schur algebra associated to H . In this paper, we discuss a relationship among the decomposition numbers of S(Λ), S(Λ) and S(Λ). In particular, we show that some important part of the decomposition matrix of S(Λ) coincides with a part of the decomposition matrix of S(Λ). §0 Introduction Let H be the Ariki-Koike algebra over an commutative integral domain R with parameters q, q, Q1, . . . , Qr ∈ R, associated to the complex reflection group Wn,r = G(r, 1, n). In [DJMa] Dipper, James and Mathas introduced the cyclotomic q-Schur algebras S(Λ) with weight poset Λ as a tool for studying the representations of the Ariki-Koike algebra H . It is an important problem to determine the decomposition matrix of S(Λ). In the case where r = 1, the cyclotomic q-Schur algebra coincides with the q-Schur algebra of [DJ]. In that case, under the condition that R = C and q is a root of unity, Varagnolo and Vasserot proved in [VV] the decomposition conjecture due to Leclerc and Thibon [LT], which provides us an algorithm of computing the decomposition matrix in connection with the canonical basis of the level 1 Fock space of type A. It is an open problem to determine the decomposition matrix for S(Λ) in the case where r ≥ 2. It is known by [A] (see also [DM]) that the determination of the decomposition matrix of S(Λ) is reduced to the case where r = 1 if the parameters satisfy the separation condition (S) qQi −Qj are invertible in R for |k| < n, i 6= j. In [U], Ugolov constructed the canonical basis of the level r Fock space of type A, and gave an algorithm of computing them. Assume that R = C, and that Qi = qi with si+1 − si ≥ n for a root of unity q ∈ C. In that case, Yvonne [Y] formulated ∗The author expresses gratitude to Professor Toshiaki Shoji. 1