Graded skew Specht modules and cuspidal modules for Khovanov-Lauda-Rouquier algebras of affine type A

TL;DR

This paper extends the construction of graded Specht modules to skew shapes in Khovanov-Lauda-Rouquier algebras of affine type A, revealing new connections with cuspidal modules and hook shapes.

Contribution

It generalizes the presentation of graded Specht modules to skew shapes and links cuspidal modules to skew Specht modules in affine type A.

Findings

Skew Specht modules can be constructed as subquotients of restrictions of Specht modules.

Cuspidal modules for affine type A are identified as skew Specht modules for hook shapes.

The construction broadens the understanding of module categories in KLR algebras.

Abstract

Kleshchev, Mathas and Ram (2012) gave a presentation for graded Specht modules over Khovanov-Lauda-Rouquier algebras of finite and affine type A. We show that this construction can be applied more generally to skew shapes to give a presentation of graded skew Specht modules, which arise as subquotients of restrictions of Specht modules. As an application, we show that cuspidal modules associated to a balanced convex preorder in affine type A are skew Specht modules for certain hook shapes.