An algorithm for semistandardising homomorphisms

TL;DR

This paper introduces a fast algorithm to express tableau homomorphisms as linear combinations of semistandard homomorphisms within the framework of Iwahori--Hecke algebra modules, enhancing computational efficiency.

Contribution

It provides a novel explicit algorithm for semistandardising homomorphisms between Specht and permutation modules, with an auxiliary result on tableau row removal.

Findings

Algorithm significantly speeds up homomorphism computations

Explicit expressions for tableau homomorphisms are obtained

Utility result simplifies tableau manipulations

Abstract

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.