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 2 n−1 (2), A 2 n (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−mδ ( m⩾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).