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

TL;DR

This paper interprets the branching coefficients of certain Lie algebra modules using Kashiwara's crystal theory, revealing new relations among Littlewood-Richardson crystals across types B, C, and D.

Contribution

It provides an explicit crystal-theoretic interpretation of branching rules for types B, C, D, and shows equivalences of LR crystals among these types in the stable region.

Findings

LR crystals of types B and D are identical to type C in the stable region.

An explicit surjection from type C LR crystal to a union of type A LR crystals is constructed.

Branching coefficients are expressed via Littlewood-Richardson coefficients in the stable region.

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.