Graded decomposition numbers of Ariki-Koike algebras for blocks of small weight

TL;DR

This paper determines the graded decomposition numbers for certain blocks of Ariki-Koike algebras of small weight, providing explicit formulas and showing some are characteristic-independent.

Contribution

It completes the description of graded decomposition numbers for blocks of weight at most two and analyzes indecomposable core blocks at level three.

Findings

Decomposition matrices are characteristic-independent for some blocks.

Provides a closed formula for graded decomposition numbers at weight two.

Analyzes indecomposable core blocks at level three.

Abstract

We present some blocks of Ariki-Koike algebras $\mathcal{H}_{n,r}$ for which the decomposition matrices are independent of the characteristic of the underlying field. We complete the description of the graded decomposition numbers for blocks of Ariki-Koike algebras of weight at most two, which consists of analysing the indecomposable core blocks at level $r = 3$, and give a closed formula for the decomposition numbers in this case.