GKM Theory for Manifolds of Isospectral Matrices in Lie Type D

TL;DR

This work extends GKM theory to isospectral matrix manifolds in Lie type D by explicitly relating the GKM graph of the skew-symmetric isospectral variety $Q_{\Gamma,\lambda}$ to that of the Hermitian counterpart $M_{\Gamma,\lambda}$. The authors construct the D-type GKM graph from the A-type graph via a doubling by $s\in(\mathbb{Z}_2)^n$ and parity-constrained edge connections, yielding a complete description of the $2^n n!$-vertex graph with weights $\epsilon_i\pm\epsilon_j$. A key result is the equivariant-formality criterion: $Q_{\Gamma,\lambda}$ is equivariantly formal if and only if $M_{\Gamma,\lambda}$ is, established through a Masuda–Panov-type argument using a real form $M'_{\Gamma,\lambda,\mathbb{R}}$ and a shared $T^1$ action. The paper confirms the Lie type D case, generalizes the known type A picture, and provides explicit cases (e.g., $\Gamma=K_2$) that illuminate the construction and its topological consequences for the cohomology of these isospectral manifolds.

Abstract

We study the manifold $Q_{Γ, λ}$ of isospectral real skew-symmetric matrices with a prescribed sparsity pattern determined by a graph $Γ$. The compact torus $T^n$ acts naturally on $Q_{Γ,λ}$ 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 $Q_{Γ, λ}$ is obtained from the GKM graph of the corresponding manifold $M_{Γ, λ}$ of isospectral Hermitian matrices. The second theorem gives a criterion for equivariant formality of $Q_{Γ, λ}$.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Definition 2.2
• Definition 2.3
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Proposition 2.8
• Definition 2.9
• Proposition 3.2