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 $U_q(\mathfrak{g})$-modules, where $\mathfrak{g}$ is $\mathfrak{so}(2n+1,\mathbb{C})$ ($B_n$-type), $\mathfrak{sp}(2n,\mathbb{C})$ ($C_n$-type), and $\mathfrak{so}(2n,\mathbb{C})$ ($D_n$-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 $C_n$ to the disjoint union of Cartesian product of LR crystals of $A_{n-1}$-type and by proving that LR crystals of types $B_n$ and $D_n$ are identical to the corresponding LR crystal of type $C_n$ in the stable region.

77 pages, proofs of several lemmas corrected

References in corpus (1)

• Super duality and Crystal bases for quantum orthosymplectic superalgebras