Construction of Irreducible Representations over Khovanov-Lauda-Rouquier Algebras of Finite Classical Type

TL;DR

This paper provides an explicit construction of irreducible modules over Khovanov-Lauda-Rouquier algebras of finite classical types using crystal basis theory, connecting module construction with crystal structures.

Contribution

It introduces a new crystal basis theoretic method to explicitly construct all irreducible modules over these algebras, linking module heads with crystal elements.

Findings

Every irreducible module is realized as the head of an induced module from certain constructed modules.

The construction is compatible with the crystal structure on $B(\infty)$ and $B(\lambda)$.

Provides an explicit, combinatorial approach to module classification over KLR algebras.

Abstract

We give an explicit construction of irreducible modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^{\lambda}$ for finite classical types using a crystal basis theoretic approach. More precisely, for each element $v$ of the crystal $B(\infty)$ (resp. $B(\lambda)$), we first construct certain modules $\nabla(\mathbf{a};k)$ labeled by the adapted string $\mathbf{a}$ of $v$. We then prove that the head of the induced module $\ind \big(\nabla(\mathbf{a};1) \boxtimes...\boxtimes \nabla(\mathbf{a};n)\big)$ is irreducible and that every irreducible $R$-module (resp. $R^{\lambda}$-module) can be realized as the irreducible head of one of the induced modules $\ind (\nabla(\mathbf{a};1) \boxtimes...\boxtimes \nabla(\mathbf{a};n))$. Moreover, we show that our construction is compatible with the crystal structure on $B(\infty)$ (resp. $B(\lambda)$).