On decomposition numbers of the cyclotomic q-Schur algebras

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