# Irreducible Modules over Khovanov-Lauda-Rouquier Algebras of type $A n$ and Semistandard Tableaux - Mathematics > Representation Theory

Irreducible Modules over Khovanov-Lauda-Rouquier Algebras of type $A n$ and Semistandard Tableaux - Mathematics > Representation Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: Using combinatorics of Young tableaux, we give an explicit construction ofirreducible graded modules over Khovanov-Lauda-Rouquier algebras $R$ and theircyclotomic quotients $R^{\lambda}$ of type $A {n}$. Our construction iscompatible with crystal structure. Let ${\mathbf B}\infty$ and ${\mathbfB}\lambda$ be the $U q\slm {n+1}$-crystal consisting of marginally largetableaux and semistandard tableaux of shape $\lambda$, respectively. On theother hand, let ${\mathfrak B}\infty$ and ${\mathfrak B}\lambda$ be the$U q\slm {n+1}$-crystals consisting of isomorphism classes of irreduciblegraded $R$-modules and $R^{\lambda}$-modules, respectively. We show that thereexist explicit crystal isomorphisms $\Phi {\infty}: {\mathbf B}\infty\overset{\sim} \longrightarrow {\mathfrak B}\infty$ and $\Phi {\lambda}:{\mathbf B}\lambda \overset{\sim} \longrightarrow {\mathfrak B}\lambda$.

Author: ** Seok-Jin Kang, Euiyong Park**

Source: https://arxiv.org/