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^λ, labelled by e-restricted partitions λ of n, for a cyclotomic KLR algebra R^Λ0_nR_n^Λ0 over a field of characteristic p ≥ 0, with mild restrictions on p. If all parts of λ are at most 2, we identify a set DStd_e,_p(λ) of standard λ-tableaux, which is defined combinatorially and naturally labels a basis of Dλ. In particular, we prove that the q-character of D^λ can be described in terms of 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.