Tensor Product of Crystal Bases for U_q(gl(m, n))-Modules

Abstract

In this paper, we give a combinatorial algorithm of decomposing the tensor product of U_q(gl(m, n))-crystals. Using the insertion scheme for (m, n)-hook semistandard tableaux, we show that there exists a bijection between the set of Littlewood– Richardson tableaux and the set of genuine highest weight vectors in the tensor product of U_q(gl(m, n))-crystals. Hence the connected components are parametrized by the Littlewood–Richardson tableaux.