辛群的倒易律和斜Pieri定则
A reciprocity law and the skew Pieri rule for the symplectic group
摘要：
我们利用斜对偶理论证明了分解辛群$Sp_{2m} = Sp_{2m}(C)$的$k$不可约表示的张量积等价于从$Sp_{2n}$分支到$Sp_{2n_1}\times\cdots\times Sp_{2n_k}$，其中$n, n_1,\ldots, n_k$是正整数，使得$n = n_1+\cdots+n_k$和$n_j$依赖于$m$以及张量积中的表示.利用这个结果和J.Lepowsky的工作，我们得到了$Sp_{2m}$的斜Pieri规则，即，一种对辛群$Sp_{2m}$的不可约表示与基本表示的张量积的不可约分解的描述。
英文摘要：
We use the theory of skew duality to show that decomposing the tensor product of $k$ irreducible representations of the symplectic group $Sp_{2m} = Sp_{2m}(C)$ is equivalent to branching from $Sp_{2n}$ to $Sp_{2n_1}\times\cdots\times Sp_{2n_k}$ where $n, n_1,\ldots, n_k$ are positive integers such that $n = n_1+\cdots+n_k$ and the $n_j$'s depend on $m$ as well as the representations in the tensor product. Using this result and a work of J. Lepowsky, we obtain a skew Pieri rule for $Sp_{2m}$, i.e., a description of the irreducible decomposition of the tensor product of an irreducible representation of the symplectic group $Sp_{2m}$ with a fundamental representation.
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