Cluster-permutohedra and submanifolds of flag varieties with torus actions

Abstract

In this paper we describe a relation between the notion of graphicahedron, introduced by Araujo-Pardo, Del R\'ıo-Francos, López-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 $Γ$. This poset is build as a combination of cosets of the symmetric group, and the geometric lattice of the graphical matroid of $Γ$. This poset is similar to the graphicahedron of $Γ$, 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 $Γ$-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.