Irreducible modules over Khovanov-Lauda-Rouquier algebras of type An 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 Rλ of type An. Our construction is compatible with crystal structure. Let B(λ) and B(λ) be the Uq(sln+1)-crystal consisting of marginally large tableaux and semistandard tableaux of shape λ, respectively. On the other hand, let B(λ) and B(λ) be the Uq(sln+1)-crystals consisting of isomorphism classes of irreducible graded R-modules and Rλ-modules, respectively. We show that there exist explicit crystal isomorphisms φ∞:B(∞)→~B(λ) and φ∞:B(λ)→~B(λ).