格点拓扑量子场论中的Mednykh公式
Mednykh's Formula via Lattice Topological Quantum Field Theories
摘要：
Mednykh证明了对任意有限群G和任意可定向曲面S，存在一个关于S的欧拉示性数线和G的不可约表示维数的#Hom（pi_1（S），G）的公式. Frobenius和Schur在不可定向的情况下证明了一个类似的公式。这两个证明都使用了特征标理论和π_1的显式表示。这些结果已用量子场论重新证明。在这里，我们提出了一个大大简化的证明，这些结果只使用初等拓扑和组合。主要的工具是一个基本不变的表面连接到一个半单代数称为格拓扑量子场论。
英文摘要：
Mednykh proved that for any finite group G and any orientable surface S, there is a formula for #Hom(pi_1(S), G) in terms of the Euler characteristic of S and the dimensions of the irreducible representations of G. A similar formula in the nonorientable case was proved by Frobenius and Schur. Both of these proofs use character theory and an explicit presentation for π_1. These results have been reproven using quantum field theory. Here we present a greatly simplified proof of these results which uses only elementary topology and combinatorics. The main tool is an elementary invariant of surfaces attached to a semisimple algebra called a lattice topological quantum field theory.
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