Cuspidal systems for affine Khovanov-Lauda-Rouquier algebras

TL;DR

This paper develops a classification framework for irreducible modules over affine Khovanov-Lauda-Rouquier algebras using cuspidal systems, introduces conjectures on modular reductions, and connects cuspidal modules to dual root vectors.

Contribution

It introduces cuspidal systems for affine KLR algebras, classifies irreducible modules, and explores their connections to dual root vectors and imaginary Schur-Weyl duality.

Findings

Classification of irreducible modules via cuspidal systems

Conjecture on reductions modulo p of irreducible modules

Identification of cuspidal modules with dual root vectors

Abstract

A cuspidal system for an affine Khovanov-Lauda-Rouquier algerba $R_\al$ yields a theory of standard modules. This allows us to classify the irreducible modules over $R_\al$ up to the so-called imaginary modules. We make a conjecture on reductions modulo $p$ of irreducible $R_\al$-modules, which generalizes James Conjecture. We also describe minuscule imaginary modules, laying the groundwork for future study of imaginary Schur-Weyl duality. We introduce colored imaginary tensor spaces and reduce a classification of imaginary modules to one color. We study the characters of cuspidal modules. We show that under the Khovanov-Lauda-Rouquier categorification, cuspidal modules correspond to dual root vectors.