A Generating Function of the Number of Homomorphisms from a Surface Group into a Finite Group

A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate the number of homomorphisms using the decomposition of the group algebra into irreducible factors. This gives a new proof of the classical formulas of Frobenius, Schur, and Mednykh. 0. Introduction Let S be a compact oriented or non-orientable surface without boundary, and χ(S) its Euler characteristic. The subject of our study is a generating function of the number |Hom(π1(S), G)| of homomorphisms from the fundamental group of S into a finite group G. We give a generating function in terms of a non-commutative integral Eqn.(2.7) or Eqn.(3.2), according to the orientability of S. The idea of such integrals comes from random matrix theory. Our integrals can be thought of as a generalization of real symmetric, complex hermitian, and quaternionic self-adjoint matrix integrals. The graphical expansion methods for real symmetric [7, 14] and complex hermitian [4] matrix integrals are generalized in [31] for quaternionic self-adjoint matrix integrals. The technique developed in [31] is further generalized in [32] to the integrals over matrices with values in non-commutative ∗-algebras. In this article we consider 1 × 1 matrix integrals over group algebras. Surprisingly, the graphical expansion of the integral gives a generating function of |Hom(π1(S), G)| for all closed surfaces. 1. Counting Formulas Computation of our generating functions Eqn.(2.7) and Eqn.(3.2) yields a new proof of the following classical counting formulas: Theorem 1.1 (Mednykh [24]). Let G be a finite group of order |G|, and Ĝ the set of all complex irreducible representations of G. By Vλ we denote the irreducible representation parameterized by λ ∈ Ĝ. Then for every compact Riemann surface S, we have