Generalised column removal for graded homomorphisms between Specht modules

Abstract

Let $n$ be a positive integer, and let $\mathscr{H}_n$ denote the affine KLR algebra in type A. Kleshchev, Mathas and Ram have given a homogeneous presentation for graded column Specht modules $\operatorname{S}_{\lambda}$ for $\mathscr{H}_n$. Given two multipartitions $\lambda$ and $\mu$, we define the notion of a \emph{dominated} homomorphism $\operatorname{S}_{\lambda}\to\operatorname{S}_{\mu}$, and use the KMR presentation to prove a generalised column removal theorem for graded dominated homomorphisms between Specht modules. In the process, we prove some useful properties of $\mathscr{H}_n$-homomorphisms between Specht modules which lead to an immediate corollary that, subject to a few demonstrably necessary conditions, every homomorphism $\operatorname{S}_{\lambda}\to\operatorname{S}_{\mu}$ is dominated, and in particular $\operatorname{Hom}_{\mathscr{H}_n}(\operatorname{S}_{\lambda},\operatorname{S}_{\mu})=0$ unless $\lambda$ dominates $\mu$. Brundan and Kleshchev show that certain cyclotomic quotients of $\mathscr{H}_n$ are isomorphic to (degenerate) cyclotomic Hecke algebras of type A. Via this isomorphism, our results can be seen as a broad generalisation of the column removal results of Fayers and Lyle and of Lyle and Mathas; generalising both into arbitrary level and into the graded setting.