Cluster-permutohedra and submanifolds of flag varieties with torus actions

TL;DR

This paper explores the relationship between cluster-permutohedra, graphicahedra, and toric topology, revealing their role in understanding the structure and homotopy properties of manifolds with torus actions, especially in the context of isospectral Hermitian matrices.

Contribution

It introduces a connection between cluster-permutohedra and graphicahedra, and applies this to analyze the topology of manifolds with torus actions, including new generalizations.

Findings

Face poset of torus action on isospectral matrices is isomorphic to cluster-permutohedron.

Homotopy properties of graphicahedra can obstruct equivariant formality.

Generalization of cluster-permutohedron describes structures of various manifolds with torus actions.

Abstract

In this paper we describe a relation between the notion of graphicahedron, introduced by Araujo-Pardo, Del R\'{\i}o-Francos, L\'{o}pez-Dudet, Oliveros, and Schulte in 2010, and toric topology of manifolds of sparse isospectral Hermitian matrices. More precisely, we recall the notion of a cluster-permutohedron, a certain finite poset defined for a simple graph $\Gamma$. This poset is build as a combination of cosets of the symmetric group, and the geometric lattice of the graphical matroid of $\Gamma$. This poset is similar to the graphicahedron of $\Gamma$, in particular, 1-skeleta of both posets are isomorphic to Cayley graphs of the symmetric group. We describe the relation between cluster-permutohedron and graphicahedron using Galois connection and the notion of a core of a finite topology. We further prove that the face poset of the natural torus action on the manifold of isospectral $\Gamma$-shaped Hermitian matrices is isomorphic to the cluster-permutohedron. Using recent results in toric topology, we show that homotopy properties of graphicahedra may serve an obstruction to equivariant formality of isospectral matrix manifolds. We introduce a generalization of a cluster-permutohedron and describe the combinatorial structure of a large family of manifolds with torus actions, including Grassmann manifolds and partial flag manifolds.