Crystal interpretation of a formula on the branching rule of types $B_{n}$, $C_{n}$, and $D_{n}$

Abstract

The branching coefficients of the tensor product of finite-dimensional irreducible Uq(g)-modules, where g is so(2n + 1, C) (Bn-type), sp(2n, C) (Cntype), and so(2n, C) (Dn-type), are expressed in terms of Littlewood-Richardson (LR) coefficients in the stable region. We give an interpretation of this relation by Kashiwara's crystal theory by providing an explicit surjection from the LR crystal of type Cn to the disjoint union of Cartesian product of LR crystals of A n−1 -type and by proving that LR crystals of types Bn and Dn are identical to the corresponding LR crystal of type Cn in the stable region.