Character Varieties of Free Groups are Gorenstein but not always Factorial

TL;DR

This paper investigates the algebraic properties of G-character varieties of free groups, establishing that they are Gorenstein universally but only factorial under certain conditions related to the group's derived subgroup.

Contribution

It proves that G-character varieties are Gorenstein in general and identifies conditions under which they are factorial, providing new insights into their algebraic structure.

Findings

X(F,G) is factorial when DG is simply connected

X(F,G) is always Gorenstein regardless of DG's properties

Counterexamples show X(F,G) need not be locally factorial when DG is not simply connected

Abstract

Fix a rank g free group F and a connected reductive complex algebraic group G. Let X(F,G) be the G-character variety of F. When the derived subgroup DG in G is simply connected we show that X(F,G) is factorial (which implies it is Gorenstein), and provide examples to show that when DG is not simply connected X(F,G) need not even be locally factorial. Despite the general failure of factoriality of these moduli spaces, using different methods, we show that X(F,G) is always Gorenstein.