Eigenvalues of matrix products

TL;DR

This paper characterizes the space of matrix pairs with prescribed eigenvalues and extends the analysis to flat $GL_n(\mathbb{C})$-structures on surfaces, providing explicit parameterizations and symplectic structures.

Contribution

It provides an explicit parameterization of matrix pairs with fixed eigenvalues and describes the moduli space of flat structures with prescribed holonomies, linking it to integrable systems.

Findings

Dimension of matrix pair space is (n-1)(n-2).

Explicit parameterization of the space of flat structures on surfaces.

Identification of a symplectic structure and integrable system related to the moduli space.

Abstract

We study pairs of matrices $A,B\in GL_n({\mathbb C})$ such that the eigenvalues of $A$, of $B$ and of the product $AB$ are specified in advance. We show that the space of such pairs $(A,B)$ under simultaneous conjugation has dimension $(n-1)(n-2)$, and give an explicit parameterization. More generally let $\Sigma$ be a surface of genus $g$ with $k$ punctures. We find a parameterization of the space $\Omega_{g,k,n}$ of flat $GL_n({\mathbb C})$-structures on $\Sigma$ whose holonomies around the punctures have prescribed eigenvalues. We show furthermore that, for $3\le k\le 2g+6$ (or $3\le k\le 9$ if $g=1$, or $3\le k$ if $g=0$), the space $\Omega_{g,k,n}$ has an explicit symplectic structure and an associated Liouville integrable system, equivalent to a leaf of a Goncharov-Kenyon dimer integrable system.