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Independence of homogeneous GKM manifolds and symmetric spaces

Shintaro Kuroki, Grigory Solomadin

TL;DR

This work develops a complete classification of independence properties for torus actions on simply connected homogeneous spaces of maximal rank. By translating isotropy weight data into weighted signed graphs tied to root systems, the authors prove that the maximal independence $k(G/H)$ lies in {2,3,n}, with $n=\dim T$, and identify when $k(G/H)=3$ or $n$ precisely with maximal rank symmetric spaces of rank $>2$, excluding a short list of cases. The analysis combines the Borel–de Siebenthal classification with graph-theoretic tools to handle classical types A–D and, separately, exceptional types F and E, yielding explicit values for each $(G,H)$. As a corollary, using Ayzenberg–Masuda results, the orbit spaces $T\backslash G/H$ are $(k+1)$-acyclic with appropriate coefficients, contributing to the topology of homogeneous GKM manifolds and addressing Masuda’s question for this class. Overall, the paper provides a detailed bridge between Lie-theoretic root data and GKM geometry, enabling precise predictions of orbit-space topology and clarifying when homogeneous GKM manifolds realize torus-manifold structures.

Abstract

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.

Independence of homogeneous GKM manifolds and symmetric spaces

TL;DR

This work develops a complete classification of independence properties for torus actions on simply connected homogeneous spaces of maximal rank. By translating isotropy weight data into weighted signed graphs tied to root systems, the authors prove that the maximal independence lies in {2,3,n}, with , and identify when or precisely with maximal rank symmetric spaces of rank , excluding a short list of cases. The analysis combines the Borel–de Siebenthal classification with graph-theoretic tools to handle classical types A–D and, separately, exceptional types F and E, yielding explicit values for each . As a corollary, using Ayzenberg–Masuda results, the orbit spaces are -acyclic with appropriate coefficients, contributing to the topology of homogeneous GKM manifolds and addressing Masuda’s question for this class. Overall, the paper provides a detailed bridge between Lie-theoretic root data and GKM geometry, enabling precise predictions of orbit-space topology and clarifying when homogeneous GKM manifolds realize torus-manifold structures.

Abstract

Let be a simply connected homogeneous space of maximal rank. Then the maximal torus -action on is a GKM manifold. We call the -action -independent if any pairwise distinct isotropy weights at a fixed point are linearly independent. Using weighted graphs, we show that the maximal independence of is , or , and that the cases of or correspond to some symmetric spaces of rank . As a corollary, using the results of Ayzenberg and Masuda, the lower-degree reduced homology groups (with appropriate coefficients) of the orbit space vanish.
Paper Structure (40 sections, 14 theorems, 39 equations, 18 figures, 4 tables)

This paper contains 40 sections, 14 theorems, 39 equations, 18 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Motivation and Main theorem
  3. Sketch of the proof
  4. Structure of the paper
  5. Preparations
  6. $k$-independence
  7. Some remarks on $k$-independence of tangential representations
  8. The torus action on the homogeneous space $G/H$
  9. The Borel-de Siebenthal theory
  10. Symmetric spaces
  11. Main theorem and its application
  12. Main theorem
  13. Proof of the main theorem
  14. An application to the topology of orbit spaces of homogeneous spaces
  15. The weighted signed graph associated from positive root systems
  16. ...and 25 more sections

Key Result

Theorem 2

Let $G/H$ be any simply connected, homogeneous space such that $T\subset H\subset G$ is a maximal torus, ${\rm rank}\ T=n$, and $G$ is a compact, connected, simple Lie group. Then one has Furthermore, $k(G/H) = 3$ or $n$ if and only if $G/H$ is a maximal rank compact symmetric space with rank $>2$ other than $B_{n}/D_{i}\times B_{n-i}$, $C_{n}/A_{n-1}\times T^{1}$, $F_{4}/C_{3}\times A_{1}$.

Figures (18)

  • Figure 1: The signed graph $K_{4}$, $\pm K_{4}$ and $K_{4}^{\circ}$ (from left). Edges with $s(e)=-$ (resp. $s(e)=+$) are drawn as solid (resp. dashed) edges.
  • Figure 2: The weighted signed graphs corresponding to several types. For the weighted signed graph $C_{4}$, the weights on the vertex can be known from the weights on loops, and edges are $a_{ij}^{-}=e_{i}-e_{j}$ and $a_{ij}^{+}=e_{i}+e_{j}$ for $1\le i<j\le 4$.
  • Figure 5:
  • Figure 6:
  • Figure 7:
  • ...and 13 more figures

Theorems & Definitions (33)

  • Theorem 2
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5: MiTo
  • Remark 2.6
  • Definition 2.7: MiTo
  • Proposition 2.8: Ba
  • Theorem 2.9
  • ...and 23 more