$B_{n}$、$C_{n}$和$D_{n}$型分支规则公式的晶体解释
Crystal interpretation of a formula on the branching rule of types $B_{n}$, $C_{n}$, and $D_{n}$
摘要：
有限维不可约$U_{q}(\mathfrak{g})$ -模的张量积的分支系数，其中$\mathfrak{g}$是$\mathfrak{so}(2n+1,\mathbb{C})$（$B_{n}$ -型），$\mathfrak{sp}(2n,\mathbb{C})$（$C_{n}$ -型）和$\mathfrak{so}(2n,\mathbb{C})$（$D_{n}$ -型），在稳定区域用Littlewood-Richardson（LR）系数表示.本文用柏原晶体理论解释了这一关系，给出了从$C_{n}$型LR晶体到$A_{n-1}$型LR晶体的笛卡尔积的不交并的一个显式满射，并证明了$B_{n}$型和$D_{n}$型LR晶体在稳定区与相应的$C_{n}$型LR晶体相同。
英文摘要：
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.
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