On decomposition numbers of the cyclotomic q-Schur algebras

Abstract

Let $S(\Lambda)$ be the cyclotomic q-Schur algebra associated to the Ariki-Koike algebra $H$. We construct a certain subalgebra $S^0(\Lambda)$ of $S(\Lambda)$, and show that it is a standardly based algebra in the sense of Du and Rui. $S^0(\Lambda)$ has a natural quotient $\bar{S^0}(\Lambda)$, which turns out to be a cellular algebra. In the case where the modified Ariki-Koike algebra $H^{\flat}$ is defined, $\bar{S^0}(\Lambda)$ coincides with the cyclotomic q-Schur algebra associated to $H^{\flat}$. In this paper, we discuss a relationship among the decomposition numbers of $S(\Lambda)$, $S^0(\Lambda)$ and $\bar{S^0}(\Lambda)$. In particular, we show that some important part of the decomposition matrix of $S(\Lambda)$ coincides with a part of the decomposition matrix of $\bar{S^0}(\Lambda)$.