Cuspidal ribbon tableaux in affine type A

TL;DR

This paper classifies and constructs cuspidal and semicuspidal skew shapes in affine type A, linking them to Specht modules and providing a combinatorial tiling framework that bounds module labels.

Contribution

It introduces a classification of cuspidal and semicuspidal skew shapes and their tilings, connecting combinatorics with representation theory in affine type A.

Findings

Cuspidal skew shapes are ribbons.

Every skew shape has a unique tiling by cuspidal ribbons.

Tiling data bounds simple module labels.

Abstract

For any convex preorder on the set of positive roots of affine type A, we classify and construct all associated cuspidal and semicuspidal skew shapes. These combinatorial objects correspond to cuspidal and semicuspidal skew Specht modules for the Khovanov-Lauda-Rouquier algebra of affine type A. Cuspidal skew shapes are ribbons, and we show that every skew shape has a unique ordered tiling by cuspidal ribbons. This tiling data provides an upper bound, in the bilexicographic order on Kostant partitions, for labels of simple factors of Specht modules.