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 ${\mathcal{H}}_n$ of the symmetric group ${\mathfrak{S}}_n$. We give an explicit fast algorithm for expressing a tableau homomorphism ${\widehat{\varphi }}_A:S^{\mu }\to M^{\lambda }$ as a linear combination of semistandard homomorphisms. Along the way we provide a utility result related to removing rows from tableaux.