Crystal bases of the Fock space representations and string functions

Abstract

Let $U_q(\mathfrak{g})$ a 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 $\mathcal{F}(\Lambda )$ be the Fock space representation for a level 1 dominant integral weight $\Lambda$. Using the crystal basis of $\mathcal{F}(\Lambda )$ and its characterization in terms of abacus, we construct an explicit bijection between the set of weight vectors in $\mathcal{F}(\Lambda {)}_{\lambda -m\delta }$ ($m\ge 0$) for a maximal weight $\lambda$ and the set of certain ordered sequences of partitions. As a corollary, we obtain the string function of the basic representation $V(\Lambda )$.

31 pages