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

Abstract

We consider simple modules for a Hecke algebra with a parameter of quantum characteristic $e$. Equivalently, we consider simple modules $D^λ$, labelled by $e$-restricted partitions $λ$ of $n$, for a cyclotomic KLR algebra $R_n^{Λ_0}$ over a field of characteristic $p\ge 0$, with mild restrictions on $p$. If all parts of $λ$ are at most $2$, we identify a set $\mathsf{DStd}_{e,p}(λ)$ of standard $λ$-tableaux, which is defined combinatorially and naturally labels a basis of $D^λ$. In particular, we prove that the $q$-character of $D^λ$ can be described in terms of $\mathsf{DStd}_{e,p}(λ)$. We show that a certain natural approach to constructing a basis of an arbitrary $D^λ$ does not work in general, giving a counterexample to a conjecture of Mathas.