Radicals of weight one blocks of Ariki-Koike algebras

TL;DR

This paper characterizes the radicals of weight one blocks in Ariki-Koike algebras, showing they are equal to a specific nilpotent ideal, and discusses applications of this finding.

Contribution

It proves that the radical of a weight one block in Ariki-Koike algebras equals a known nilpotent ideal, extending understanding of their structure.

Findings

Radical of weight one block equals a specific nilpotent ideal.

Provides applications of the radical characterization.

Enhances structural understanding of Ariki-Koike algebra blocks.

Abstract

Let $K$ be a field and $q\in K$, $q\neq 0, 1$. 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 $r\geq 2$, $Q_i=q^{a_i}$, $a_i\in \mathbb{Z}$. 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.