On bases of some simple modules of symmetric groups and Hecke algebras

TL;DR

This paper studies simple modules of Hecke algebras and cyclotomic KLR algebras, identifying bases for certain modules and describing their characters, while also providing counterexamples to existing conjectures.

Contribution

It introduces a combinatorial basis for simple modules labeled by partitions with parts at most two and describes their $q$-characters, challenging previous assumptions.

Findings

Identified a natural basis for modules with partitions of parts at most two.

Described the $q$-character of these modules using combinatorial tableaux.

Provided a counterexample to a conjecture by Mathas.

Abstract

We consider simple modules for a Hecke algebra with a parameter of quantum characteristic $e$. Equivalently, we consider simple modules $D^{\lambda}$, labelled by $e$-restricted partitions $\lambda$ of $n$, for a cyclotomic KLR algebra $R_n^{\Lambda_0}$ over a field of characteristic $p\ge 0$, with mild restrictions on $p$. If all parts of $\lambda$ are at most $2$, we identify a set $\mathsf{DStd}_{e,p}(\lambda)$ of standard $\lambda$-tableaux, which is defined combinatorially and naturally labels a basis of $D^{\lambda}$. In particular, we prove that the $q$-character of $D^{\lambda}$ can be described in terms of $\mathsf{DStd}_{e,p}(\lambda)$. We show that a certain natural approach to constructing a basis of an arbitrary $D^{\lambda}$ does not work in general, giving a counterexample to a conjecture of Mathas.