Irreducible Modules over Khovanov-Lauda-Rouquier Algebras of type $A_n$ and Semistandard Tableaux

TL;DR

This paper constructs explicit combinatorial models for irreducible modules over Khovanov-Lauda-Rouquier algebras of type A using Young tableaux, establishing crystal isomorphisms with known combinatorial crystals.

Contribution

It provides an explicit combinatorial construction of irreducible modules and demonstrates their compatibility with crystal structures, linking algebraic modules to Young tableaux.

Findings

Explicit crystal isomorphisms between module classes and tableaux crystals

Construction compatible with crystal structure

Connection between algebraic modules and combinatorial tableaux

Abstract

Using combinatorics of Young tableaux, we give an explicit construction of irreducible graded modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^{\lambda}$ of type $A_{n}$. Our construction is compatible with crystal structure. Let ${\mathbf B}(\infty)$ and ${\mathbf B}(\lambda)$ be the $U_q(\slm_{n+1})$-crystal consisting of marginally large tableaux and semistandard tableaux of shape $\lambda$, respectively. On the other hand, let ${\mathfrak B}(\infty)$ and ${\mathfrak B}(\lambda)$ be the $U_q(\slm_{n+1})$-crystals consisting of isomorphism classes of irreducible graded $R$-modules and $R^{\lambda}$-modules, respectively. We show that there exist explicit crystal isomorphisms $\Phi_{\infty}: {\mathbf B}(\infty) \overset{\sim} \longrightarrow {\mathfrak B}(\infty)$ and $\Phi_{\lambda}: {\mathbf B}(\lambda) \overset{\sim} \longrightarrow {\mathfrak B}(\lambda)$.