A skew Specht perspective of RoCK blocks and cuspidal systems for KLR algebras in affine type A

TL;DR

This paper introduces a new combinatorial approach using skew diagrams and Specht modules to understand simple modules in affine type A KLR algebras, linking cuspidal systems and RoCK blocks.

Contribution

It constructs explicit skew diagrams for cuspidal modules and develops core-truncation functors connecting RoCK blocks to imaginary semicuspidal modules.

Findings

Explicit skew diagrams $()$ for simple modules.

Development of core-truncation functors.

Characterization of core and RoCK blocks via cuspidal tilings.

Abstract

Cuspidal systems parameterize KLR algebra representations via root partitions $\pi$, where simple modules $L(\pi)$ arise as heads of proper standard modules. Working in affine type A with an arbitrary convex preorder, we construct explicit skew diagrams $\zeta(\pi)$ such that the skew Specht module $S^{\zeta(\pi)}$ has simple head $L(\pi)$ and a filtration by proper standard modules. A key ingredient in this construction is the development of `core-truncation' functors, which take module categories of level one RoCK blocks to the category of imaginary semicuspidal KLR modules. Every simple imaginary semicuspidal module arises in the image of these functors. This result stems from an in-depth study of the combinatorial interplay between cuspidal systems and RoCK cyclotomic KLR algebras, in which we characterize core blocks and RoCK blocks in arbitrary level via cuspidal tiling properties of multipartitions in these blocks.