On homomorphisms indexed by semistandard tableaux

We study the homomorphism spaces between Specht modules for the Hecke algebras $\h$ of type $A$. We prove a cellular analogue of the kernel intersection theorem and a $q$-analogue of a theorem of Fayers and Martin and apply these results to give an algorithm which computes the homomorphism spaces $\Hom_{\h}(S^μ,S^λ)$ for certain pairs of partitions $λ$ and $μ$. We give an explicit description of the homomorphism spaces $\Hom_\h(S^μ,S^λ)$ where $\h$ is an algebra over the complex numbers, $λ=(λ_1,λ_2)$ and $μ$ is an arbitrary partition with $μ_1 \geq λ_2$.

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