Irreducible Specht modules for Hecke algebras of type A

Let F be a field, n a non-negative integer, λ a partition of n and S^λ the corresponding Specht module for the Iwahori-Hecke algebra H_F,q(S_n). James and Mathas conjecture a necessary and sufficient condition on λ for S^λ to be irreducible. We prove the sufficiency of this condition in the case where F has infinite characteristic and also in the case where q=1.