Cyclotomic $q$-Schur algebras associated to the Ariki-Koike algebra

TL;DR

This paper introduces subalgebras and quotient algebras of cyclotomic q-Schur algebras related to Ariki-Koike algebras, revealing how their decomposition numbers factor into those of smaller associated algebras.

Contribution

It defines new subalgebras and quotient algebras of cyclotomic q-Schur algebras and demonstrates their role in factorizing decomposition numbers.

Findings

Subalgebras $S^p$ are standardly based.

Quotient algebras $ar S^p$ are cellular.

Decomposition numbers factor into smaller Ariki-Koike algebra components.

Abstract

Let $S$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $H_{n,r}$ of rank $n$, introduced by Dipper-James-Mathas. For each $p = (r_1, ..., r_g)$ such that $r_1 + ... + r_g = r$, we define a subalgebra $S^p$ of $S$ and its quotient algebra $\bar S^p$. It is shown that $S^p$ is a standardly based algebra and $\bar S^p$ is a cellular algebra. By making use of these algebras, we show that certain decomposition numbers for $S$ can be expressed as a product of decomposition numbers for cyclotomic $q$-Schur algebras associated to smaller Ariki_koike algebras $H_{n_k,r_k}$.