Centralizers of irreducible subgroups in the projective special linear group

TL;DR

This paper classifies conjugacy classes of centralizers of irreducible subgroups in PSL(n,C) using finite abelian groups with bilinear forms, and explores their role in understanding the structure of character varieties.

Contribution

It introduces a classification method for these centralizers via alternate modules and establishes a finite graph framework for their combinatorial analysis.

Findings

Classification of conjugacy classes for squarefree n

Finite graph structure related to centralizer classification

Insights into the stratification of singular loci in character varieties

Abstract

In this paper, we classify conjugacy classes of centralizers of irreducible subgroups in $PSL(n,\mathbb{C})$ using alternate modules a.k.a. finite abelian groups with an alternate bilinear form. When $n$ is squarefree, we prove that these conjugacy classes are classified by their isomorphism classes. More generally, we define a finite graph related to this classification whose combinatorial properties are expected to help us describe the stratification of the singular (orbifold) locus in some character varieties.