Invariants of pairs in SL(4,C) and SU(3,1)

TL;DR

This paper develops a minimal coordinate system for character varieties of free groups in SL(4,C) and SU(3,1), enabling a better understanding of conjugation classes of matrix pairs in these groups.

Contribution

It introduces a minimal global coordinate system for SL(4,C) character varieties and simplifies the description of conjugation classes in SU(3,1) using symmetry and relations.

Findings

Constructed a 30-coordinate system for SL(4,C) character variety.

Reduced to 22 coordinates with 5 real relations for SU(3,1).

Provided a framework for classifying generic matrix pairs.

Abstract

We describe a minimal global coordinate system of order 30 on the SL(4,C)-character variety of a rank 2 free group. Using symmetry within this system, we obtain a smaller collection of 22 coordinates subject to 5 further real relations that determine conjugation classes of generic pairs of matrices in SU(3,1).