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

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.