关于$Fl_n \times Fl_n \times Fl_n //SU(n)$和$SU(n)$张量积重数的可积系统
Integrable systems on $Fl_n \times Fl_n \times Fl_n //SU(n)$ and $SU(n)$ tensor product multiplicities
摘要：
我们构造了一个稠密定义的环面作用于三个完全旗簇乘积的辛商。相应的矩映射的像的闭包是一个凸多胞体。这个空间的几何量子化的维度给出了$SU(n)$的表示环的结构常数，我们通过计算矩映射的图像中的格点来给出。Berenstein-Zelevinsky给出了这种结构常数的格点公式;我们的结果显示了这种公式是如何作为“偏振不变性”的一个例子在几何上出现的，类似于Guillemin-Sternberg对Gelfand-Cetlin多面体的描述。我们概述了循环群和向量丛的模空间的应用。
英文摘要：
We construct a densely defined torus action on the symplectic quotient of the product of three complete flag varieties. The closure of the image of the corresponding moment map is a convex polytope. The dimension of the geometric quantization of this space gives the structure constants of the representation ring of $SU(n)$, which we show is given by counting lattice points in the image of the moment map. Such lattice point formulas for the structure constants were given by Berenstein-Zelevinsky; our results show how such formulas arise geometrically as an example of `invariance of polarization', analogous to the description of the Gelfand-Cetlin polytopes by Guillemin-Sternberg. We outline applications to loop groups and to moduli spaces of vector bundles.
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