Generalised column removal for graded homomorphisms between Specht modules

Abstract

Let $n$ be a positive integer, and let ${\mathcal{H}}_n$ denote the affine KLR algebra in type A. Kleshchev, Mathas and Ram have given a homogeneous presentation for graded column Specht modules $S_{\lambda }$ for ${\mathcal{H}}_n$. Given two multipartitions $\lambda$ and $\mu$, we define the notion of a \emph{dominated} homomorphism $S_{\lambda }\to S_{\mu }$, 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 ${\mathcal{H}}_n$-homomorphisms between Specht modules which lead to an immediate corollary that, subject to a few demonstrably necessary conditions, every homomorphism $S_{\lambda }\to S_{\mu }$ is dominated, and in particular ${\mathrm{Hom}}_{{\mathcal{H}}_n}(S_{\lambda },S_{\mu })=0$ unless $\lambda$ dominates $\mu$. Brundan and Kleshchev show that certain cyclotomic quotients of ${\mathcal{H}}_n$ 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.

34 pages

References in corpus (2)

• 2-Kac-Moody algebras
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

Cited by in corpus (8)

• The many graded cellular bases of Hecke algebras
• On the semisimplicity of the cyclotomic quiver Hecke algebra of type C
• On bases of some simple modules of symmetric groups and Hecke algebras
• Specht modules labelled by hook bipartitions I
• Modular decomposition numbers of cyclotomic Hecke and diagrammatic Cherednik algebras: A path theoretic approach
• An analogue of row removal for diagrammatic Cherednik algebras
• Decomposable Specht modules indexed by bihooks
• Decomposable Specht modules indexed by bihooks II