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

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.