GKM Theory for Manifolds of Isospectral Matrices in Lie Type D

TL;DR

This paper explores the structure of manifolds of isospectral skew-symmetric matrices with specific sparsity patterns, applying GKM theory to understand their GKM graphs and conditions for equivariant formality.

Contribution

It establishes how the GKM graph of skew-symmetric matrices relates to that of Hermitian matrices and provides criteria for their equivariant formality, advancing the understanding of these manifolds in Lie type D.

Findings

GKM graph of skew-symmetric matrices derived from Hermitian case

Criterion for equivariant formality of the manifold

Connection between matrix spectra and GKM theory

Abstract

We study the manifold $Q_{\Gamma, \lambda}$ of isospectral real skew-symmetric matrices with a prescribed sparsity pattern determined by a graph $\Gamma$. The compact torus $T^n$ acts naturally on $Q_{\Gamma,\lambda}$ by conjugation, and this action can be studied using GKM theory. We prove two results about this manifold and its GKM graph. The first theorem describes how the GKM graph of $Q_{\Gamma, \lambda}$ is obtained from the GKM graph of the corresponding manifold $M_{\Gamma, \lambda}$ of isospectral Hermitian matrices. The second theorem gives a criterion for equivariant formality of $Q_{\Gamma, \lambda}$.