Radicals of weight one blocks of Ariki-Koike algebras

Abstract

Let $K$ be a field and $q\in K$, $q\ne 0,1$. Let ${\mathcal{H}}_n(q,Q)$ be an Ariki-Koike algebra, where the cyclotomic parameter $Q=(Q_1,Q_2,\cdots ,Q_r)\in K^r$ with $r\ge 2$, $Q_i=q^{a_i}$, $a_i\in \mathbb{Z}$. For a weight one block $B$ of ${\mathcal{H}}_n(q,Q)$, we prove in this paper that $\text{rad}B=I$, where $I$ is the nilpotent ideal constructed for a symmetric cellular algebra in [Radicals of symmetric cellular algebras, Colloq. Math. {\bf 133} (2013) 67-83]. We also give some applications of this result.

9 pages