Generalized Zeta function representation of groups and 2-dimensional Topological Yang-Mills theory: The example of GL(2, F_q) and PGL(2, F_q)

TL;DR

This paper explores generalized Zeta function representations of groups and their connection to 2D topological Yang-Mills theory, providing new formulas, computations for specific groups, and insights into their representation theory.

Contribution

It introduces generalized Zeta functions for groups, extends Mednikh formulas, and applies these to finite groups like GL(2, F_q) and PGL(2, F_q).

Findings

Derived new formulas relating Zeta functions and Yang-Mills theory.

Computed Zeta functions and character tables for GL(2, F_q) and PGL(2, F_q).

Analyzed the fusion ring structures and Frobenius-Schur indicators of these groups.

Abstract

We recall the relation between Zeta function representation of groups and two-dimensional topological Yang-Mills theory through Mednikh formula. We prove various generalisations of Mednikh formulas and define generalization of Zeta functions representations of groups. We compute some of these functions in the case of the finite group $GL(2, {\mathbb F}_q)$ and $PGL(2,{\mathbb F}_q).$ We recall the table characters of these groups for any $q$, compute the Frobenius-Schur indicator of their irreducible representations and give the explicit structure of their fusion rings