Generating sets for coordinate rings of character varieties

TL;DR

This paper identifies finite generator sets for the coordinate rings of G-character varieties across classical and exceptional groups, enabling explicit polynomial descriptions and advancing understanding of their algebraic structure.

Contribution

It provides the first comprehensive finite generator sets for coordinate rings of G-character varieties for all classical and exceptional groups, along with an algorithm for explicit polynomial equations.

Findings

Finite generator sets for classical groups' character varieties.

Finite generator sets for fields of rational functions of exceptional groups.

Algorithm for explicit polynomial descriptions of character varieties.

Abstract

We find finite, reasonably small, generator sets of the coordinate rings of G-character varieties of finitely generated groups for all classical groups G. This result together with the method of Grobner basis gives an algorithm for describing character varieties by explicit polynomial equations. Additionally, we describe finite sets of generators of the fields of rational functions on G-character varieties for all exceptional algebraic groups G.