On homomorphisms indexed by semistandard tableaux

We describe techniques which may be used to compute the homomorphism space between Specht modules for the Hecke algebras of type A. We prove a q-analogue of a result of Fayers and Martin and show how it may be applied to construct homomorphisms between Specht modules. In particular, we show that in certain cases the dimension of the homomorphism space is given by the corank of a matrix whose entries we write down explicitly.