Fourier-Deligne变换与对称群的表示
Fourier-Deligne transform and representations of the symmetric group
摘要：
我们计算了局部系统${\mathcal L}_Λ$在$\Conf_n(¶^1)$上的圆锥上的IC延拓到${\C}^{n+1}$的Fourier-Deligne变换，该圆锥与$S_n$的表示$Λ$相关联，其中$Λ$的Young图的第一行的长度$n-k$至少为$\frac{|Λ|-1}{2}$。答案是IC扩展到对偶向量空间${\C}^{n+1}$的本地系统${\mathcal R}_λ$上的$¶^n$，其中${\mathcal R}_λ$对应于$S_k$的表示$λ$，其杨氏图是从$Λ$的杨氏图中删除其第一行得到的。我们还证明了一个类似的陈述$S_n$ -局部系统的Abel-Jacobi映射的纤维。我们使用我们的结果上的傅里叶-德利涅变换重新推导的一部分结果的Michel Brion的克罗内克系数。
英文摘要：
We calculate the Fourier-Deligne transform of the IC extension to ${\C}^{n+1}$ of the local system ${\mathcal L}_Λ$ on the cone over $\Conf_n(¶^1)$ associated to a representation $Λ$ of $S_n$, where the length $n-k$ of the first row of the Young diagram of $Λ$ is at least $\frac{|Λ|-1}{2}$. The answer is the IC extension to the dual vector space ${\C}^{n+1}$ of the local system ${\mathcal R}_λ$ on the cone over the $k$-th secant variety of the rational normal curve in $¶^n$, where ${\mathcal R}_λ$ corresponds to the representation $λ$ of $S_k$, the Young diagram of which is obtained from the Young diagram of $Λ$ by deleting its first row. We also prove an analogous statement for $S_n$-local systems on fibers of the Abel-Jacobi map. We use our result on the Fourier-Deligne transform to rederive a part of a result of Michel Brion on Kronecker coefficients.
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