An algorithm for semistandardising homomorphisms

Suppose $\mu$ is a partition of $n$ and $\lambda$ a composition of $n$, and let $S^\mu$, $M^\lambda$ denote the Specht module and permutation module defined by Dipper and James for the Iwahori--Hecke algebra $\mathscr{H}_n$ of the symmetric group $\mathfrak{S}_n$. We give an explicit fast algorithm for expressing a tableau homomorphism $\hat\phi_A:S^\mu\to M^\lambda$ as a linear combination of semistandard homomorphisms. Along the way we provide a utility result related to removing rows from tableaux.