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(∞) 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(∞) 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 and .