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 $\overline{S^0}(\Lambda )$, which turns out to be a cellular algebra. In the case where the modified Ariki-Koike algebra $H^{♭}$ is defined, $\overline{S^0}(\Lambda )$ coincides with the cyclotomic q-Schur algebra associated to $H^{♭}$. In this paper, we discuss a relationship among the decomposition numbers of $S(\Lambda )$, $S^0(\Lambda )$ and $\overline{S^0}(\Lambda )$. In particular, we show that some important part of the decomposition matrix of $S(\Lambda )$ coincides with a part of the decomposition matrix of $\overline{S^0}(\Lambda )$.