非交换矩阵积分与有限群中曲面群的表示簇
Non-commutative matrix integrals and representation varieties of surface groups in a finite group
摘要：
利用带状图及其不可定向对应物Moebius图，得到了有限维实或复von Neumann代数中非交换矩阵积分的一个图形展开式.每个图的贡献是绘制该图的表面的拓扑型的不变量。作为例子，我们计算了有限群的群代数上的积分。证明了该积分是从任意闭曲面的基本群到有限群的同态个数的母函数。图形的扩展公式产生一个新的证明的经典定理Frobenius，舒尔和Mednykh对这些号码。
英文摘要：
A graphical expansion formula for non-commutative matrix integrals with values in a finite-dimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their non-orientable counterpart called Moebius graphs. The contribution of each graph is an invariant of the topological type of the surface on which the graph is drawn. As an example, we calculate the integral on the group algebra of a finite group. We show that the integral is a generating function of the number of homomorphisms from the fundamental group of an arbitrary closed surface into the finite group. The graphical expansion formula yields a new proof of the classical theorems of Frobenius, Schur and Mednykh on these numbers.
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