2024/07/15 by Kenyon, Richard, Ovenhouse, Nicholas
#05E10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2407.10786
We study pairs of matrices A,B∈ GLn(\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 Σ be a surface of genus g with k punctures. We find a parameterization of the space Ωg,k,n of flat GLn(\mathbb C)-structures on Σ whose holonomies around the punctures have prescribed eigenvalues. We show furthermore that, for 3≤ k≤ 2g+6 (or 3≤ k≤ 9 if g=1, or 3≤ k if g=0), the space Ωg,k,n has an explicit symplectic structure and an associated Liouville integrable system, equivalent to a leaf of a Goncharov-Kenyon dimer integrable system.