GKM Theory for Manifolds of Isospectral Matrices in Lie Type D

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 $Q_{Γ, λ}$.

💡 Research Summary

The paper investigates the manifold (Q_{\Gamma,\lambda}) consisting of real skew‑symmetric matrices whose non‑zero entries are prescribed by a simple connected graph (\Gamma) and whose eigenvalues are the fixed generic set (\lambda=(\lambda_{1},\dots,\lambda_{n})). The compact torus (T^{n}) acts on this manifold by conjugation, and the authors study this action through the lens of GKM (Goresky‑Kottwitz‑MacPherson) theory.

First, the authors recall the construction of the space of all real skew‑symmetric matrices (Q_{2n}) and view each matrix as an (n\times n) block matrix with (2\times2) blocks. A matrix is called (\Gamma)-shaped if the block in position ((i,j)) is zero whenever the edge ({i,j}) does not belong to the edge set (E(\Gamma)). The subspace (Q_{\Gamma}) of all such matrices has dimension (n+4|E(\Gamma)|). Imposing the spectrum condition yields the submanifold (Q_{\Gamma,\lambda}). For a generic choice of (\lambda) (pairwise distinct, non‑zero) the Pfaffian of a skew‑symmetric matrix is non‑zero, which splits (Q_{\Gamma,\lambda}) into two diffeomorphic connected components (Q^{+}{\Gamma,\lambda}) and (Q^{-}{\Gamma,\lambda}). Each component has real dimension (4|E(\Gamma)|).

The torus (T^{n}) is realized as the diagonal subgroup of (SO(2n)) (or equivalently as ({ \operatorname{diag}(t_{1},\dots,t_{n})\mid |t_{i}|=1})). Its conjugation action preserves the block structure, the spectrum, and the Pfaffian sign, thus it restricts to an effective action on each component of (Q_{\Gamma,\lambda}). The fixed points of this action are described explicitly: they are obtained by first permuting the eigenvalues via a permutation (\sigma\in\mathfrak{S}{n}) (giving the diagonal Hermitian fixed points of the related Hermitian manifold (M{\Gamma,\lambda})) and then assigning a sign vector (s\in{\pm1}^{n}) to the (2\times2) blocks. Consequently the fixed‑point set is in bijection with (\mathfrak{S}{n}\times\mathbb{Z}{2}^{n}).

The paper then recalls the basic notions of torus actions, Borel constructions, equivariant formality, and GKM graphs. A GKM manifold is a compact (T)-manifold with isolated fixed points such that the one‑skeleton is a union of invariant two‑spheres; equivariant formality means that the Serre spectral sequence of the Borel fibration collapses at (E_{2}).

The first main result (Theorem 1.1) gives a concrete combinatorial recipe for the GKM graph of (Q_{\Gamma,\lambda}) in terms of the already known GKM graph of the Hermitian isospectral manifold (M_{\Gamma,\lambda}). The authors introduce Construction 4.1: starting from the GKM graph (\mu) of (M_{\Gamma,\lambda}), each vertex (\sigma) is replaced by (2^{n}) vertices labeled (A_{s,\sigma}) (one for each sign vector). For each edge of (\mu) connecting (\sigma) and (\sigma\circ(i,j)), the construction creates (2^{n}) edges in the new graph, each joining a pair of vertices whose sign vectors differ only in the (i) and (j) coordinates in a prescribed way (either swapping the signs or flipping both). The axial function on these new edges is defined to be (\varepsilon_{i}\pm\varepsilon_{j}), where (\varepsilon_{k}) denotes the standard basis of (\operatorname{Hom}(T^{n},S^{1})). This procedure yields a graph (\eta) that is precisely the GKM graph of (Q_{\Gamma,\lambda}). The authors verify that the number of vertices becomes (n!\cdot2^{n}) and the number of edges becomes (|E(\Gamma)|\cdot n!\cdot2^{n-1}), matching the expected dimension of the one‑skeleton. Proposition 4.6 describes the Hermitian graph, while Proposition 4.11 establishes the corresponding description for the skew‑symmetric case, completing the proof of Theorem 1.1.

The second main result (Theorem 1.2) establishes an equivalence of equivariant formality between the two manifolds: (Q_{\Gamma,\lambda}) is equivariantly formal if and only if (M_{\Gamma,\lambda}) is. The proof relies on a theorem of Masuda and Panov (