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 article that $\text{rad}B=I,$ where I is the nilpotent ideal constructed for a symmetric cellular algebra [Radicals of symmetric cellular algebras, Colloq. Math. 133 (2013) 67–83]. We also give some applications of this result.

Keywords:

• Ariki–Koike algebra
• block
• radical
• weight

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16G30
• 16N20

Acknowledgments

Authors are sincerely grateful to the anonymous referee for the careful reading and valuable comments, especially for pointing out an error in a preliminary version of this paper. Yanbo Li would like to express his sincere thanks to Chern Institute of Mathematics of Nankai University for the hospitality during his visit.

Additional information

Funding

Li is supported by the Natural Science Foundation of Hebei Province, China (A2017501003) and NSFC 11871107.