Spectral data for pairs of matrices of order 3 and action of the group $GL(2,Z)$

TL;DR

This paper provides explicit formulas describing how the group $GL(2,Z)$ acts on pairs of 3x3 matrices using spectral data, including spectral curves and line bundles, enhancing understanding of their symmetries.

Contribution

It introduces explicit formulas for the action of $GL(2,Z)$ on pairs of 3x3 matrices via spectral data, which was previously not explicitly characterized.

Findings

Derived formulas for the action of $GL(2,Z)$ on spectral data of matrix pairs.

Connected group actions with geometric objects like spectral curves and line bundles.

Facilitated analysis of matrix pairs through spectral invariants.

Abstract

The group $GL(2,Z)$ acts in a natural way on the set of pairs of $n\times n$-matrices determined up to a simultaneous conjugation. For $n=3$ we write explicit formulas for action of generators of $GL(2,Z)$ in the terms of spectral data of matrices, i.e., spectral curves and line bundles.