Representations of Khovanov-Lauda-Rouquier Algebras and Combinatorics of Lyndon Words

TL;DR

This paper constructs irreducible representations of affine Khovanov-Lauda-Rouquier algebras of finite type, linking highest weights to Lyndon words and confirming predictions by Leclerc.

Contribution

It provides a new construction of irreducible modules for affine KLR algebras using combinatorics of Lyndon words, extending previous theories.

Findings

Irreducible modules are simple heads of induced modules.

Highest weights correspond to good words and Lyndon words.

Construction aligns with Leclerc's predictions.

Abstract

We construct irreducible representations of affine Khovanov-Lauda-Rouquier algebras of arbitrary finite type. The irreducible representations arise as simple heads of appropriate induced modules, and thus our construction is similar to that of Bernstein and Zelevinsky for affine Hecke algebras of type A. The highest weights of irreducible modules are given by the so-called good words, and the highest weights of the 'cuspidal modules' are given by the good Lyndon words. In a sense, this has been predicted by Leclerc.