Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Counting homomorphisms from surface groups to finite groups

Michael R. Klug

Abstract

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group $G$, where conjugacy classes of the boundary components of the surface must map to prescribed conjugacy classes in $G$, to a sum over values of irreducible characters of $G$ weighted by Frobenius-Schur multipliers. The proof is structured so that the corresponding results for closed and possibly orientable surfaces, as well as some generalizations, are derived using the same methods. We then apply these results to the specific case of the symmetric group.

Counting homomorphisms from surface groups to finite groups

Abstract

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group , where conjugacy classes of the boundary components of the surface must map to prescribed conjugacy classes in , to a sum over values of irreducible characters of weighted by Frobenius-Schur multipliers. The proof is structured so that the corresponding results for closed and possibly orientable surfaces, as well as some generalizations, are derived using the same methods. We then apply these results to the specific case of the symmetric group.

Paper Structure

This paper contains 4 sections, 12 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Counting solutions to equations in groups
  3. Relationship with 2-dimensional topology
  4. Some symmetric function identities

Key Result

Theorem 1

(Frobenius, Schur) Let $G$ be a finite group and $w \in G$ an element. Then and where the sums are over the irreducible characters of $G$.

Figures (1)

  • Figure 1: This figure shows the nonorientable compact surface $N_{n,k}$ of nonorientable genus $k$ with $n$ boundary components. The picture is of a sphere (drawn as just the plane) with $k$ circles where the disks that these circles bound are removed (the lower circles) and $n$ circles (those marked with an X) where the disks that these circles bound have been removed and replaced with Möbius bands. The boundary components are the lower circles and they have been given a specific orientation.

Theorems & Definitions (17)

  • Theorem 1
  • Proposition 1
  • Corollary 1
  • Remark 1
  • Proposition 2
  • Proposition 3
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 7 more