Construction of irreducible representations over Khovanov-Lauda-Rouquier algebras of finite classical type

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