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 $\Phi_{ \lambda}: { \mathbf B}(\lambda) \overset{ \sim} \longrightarrow { \mathfrak B}(\lambda)$.
Categories: math.RT