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

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. $B(\lambda )$).

Version 3: We change the notation \nabla by Δ, put one remark on Δ's, and update references