Crystal bases of the Fock space representations and string functions

Abstract

Let U_q(g) be the quantum affine algebra of type A_n^(1), A_2n−1^(2), A_2n^(2), B_n^(1), D_n^(1), and D_n+1^(2), and let F(Λ) be the Fock space representation for a level 1 dominant integral weight Λ. Using the crystal basis of F(Λ) and its characterization in terms of abacus, we construct an explicit bijection between the set of weight vectors in F(Λ)_{λ-m \delta} (m \geq 0) for a maximal weight λ and the set of certain ordered sequences of partitions. As a corollary, we obtain the string function of the basic representation V(Λ).