Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux

Abstract

In this paper, we give an explicit combinatorial realization of the crystal B ( λ ) B(\lambda ) for an irreducible highest weight U q ( q ( n ) ) U_q(\mathfrak {q}(n)) -module V ( λ ) V(\lambda ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms for decomposing the tensor product of q ( n ) \mathfrak {q}(n) -crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.