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

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