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

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