Independence of homogeneous GKM manifolds and symmetric spaces

Let $G/H$ be a simply connected homogeneous space of maximal rank. Then the maximal torus $T$-action on $G/H$ is a GKM manifold. We call the $T$-action $j$-independent if any $i(\leq j)$ pairwise distinct isotropy weights at a fixed point are linearly independent. Using weighted graphs, we show that the maximal independence of $G/H$ is $2$, $3$ or $n=\dim T$, and that the cases of $3$ or $n=\dim T$ correspond to some symmetric spaces of rank $>2$. As a corollary, using the results of Ayzenberg and Masuda, the lower-degree reduced homology groups (with appropriate coefficients) of the orbit space $T\backslash G/H$ vanish.

💡 Research Summary

The paper investigates the independence properties of torus actions on homogeneous spaces of maximal rank, focusing on the class of GKM (Goresky‑Kottwitz‑MacPherson) manifolds. For a compact, connected, simple Lie group (G) and a closed subgroup (H) containing a maximal torus (T) (so that (\operatorname{rank}T=\operatorname{rank}H=\operatorname{rank}G=n)), the quotient (G/H) is simply‑connected, even‑dimensional, and its (T)‑action has isolated fixed points. At each fixed point the isotropy representation decomposes into one‑dimensional weight spaces whose weights are precisely the positive roots of (G) that are not roots of (H); i.e. the set (\Delta_{G,H}=\Delta_G\setminus\Delta_H).

The authors introduce the notion of (j)-independence for a torus action: a (T)‑manifold (M) is (j)-independent if, at every fixed point, any collection of at most (j) distinct isotropy weights is linearly independent. The maximal such integer is denoted (k(M)). For a GKM manifold, (k(M)\ge2) by definition; higher values indicate stronger combinatorial rigidity and are closely related to torus manifolds (the case (k(M)=\dim T)).

The central problem is to compute (k(G/H)) for all homogeneous spaces of maximal rank. The key observation is that (k(G/H)) depends only on the combinatorics of the root complement (\Delta_{G,H}). This set can be viewed as the ground set of a matroid whose circuits correspond to linearly dependent subsets of weights. Detecting circuits is equivalent to checking linear dependencies among roots, a task that can be encoded in a weighted signed graph: vertices represent roots, edges encode the relation that the sum of two roots is again a root, and signs distinguish positive from negative roots. A circuit appears as a cycle in this graph.

To analyze (\Delta_{G,H}) systematically, the authors employ Borel‑de Siebenthal theory, which classifies maximal‑rank subgroups (H) of a simple Lie group (G) via operations on the extended Dynkin diagram of (G). Deleting a node (or halving a circle subgroup) corresponds to removing a set of simple roots, thereby determining (\Delta_H). This yields an explicit description of (\Delta_{G,H}) for every pair ((G,H)).

The paper treats separately the classical families (types A, B, C, D) and the exceptional groups (F₄, E₆, E₇, E₈). For the classical types, the complement matroid can be analyzed directly using the weighted signed graph technique. The authors compute circuits and find that, except for a few special cases, the matroid has circuits of size three, which forces (k(G/H)=2). The exceptional cases are handled by a case‑by‑case inspection, again confirming the same pattern.

The main theorem (Theorem 2) states that for any simply‑connected homogeneous space of maximal rank, \