Irreducible Modules over Khovanov-Lauda-Rouquier Algebras of type $A_n$ and Semistandard 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 ${\Phi }_{\infty }:\mathbf{B}(\infty )\stackrel{\sim }{⟶}\mathfrak{B}(\infty )$ and ${\Phi }_{\lambda }:\mathbf{B}(\lambda )\stackrel{\sim }{⟶}\mathfrak{B}(\lambda )$.