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 over a field of characteristic p ae<yen> 0, with mild restrictions on p. If all parts of lambda are at most 2, we identify a set 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 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.