Poset topology, moves, and Bruhat interval polytope lattices

TL;DR

This paper investigates the topology of lattices derived from polytope orientations, revealing conditions under which their order complexes are homotopy equivalent to spheres or contractible, and explores chain connectivity and implications for plabic graphs.

Contribution

It generalizes known results on permutahedra to Bruhat interval polytopes and establishes homotopy and connectivity properties of related lattices.

Findings

Order complex of an interval is homotopy equivalent to a sphere if the corresponding face exists.

Saturated chains are highly connected via moves across 2-faces.

Results extend to Grassmannian permutations, impacting plabic graph theory.

Abstract

We study the poset topology of lattices arising from orientations of 1-skeleta of directionally simple polytopes, with Bruhat interval polytopes $Q_{e,w}$ as our main example. We show that the order complex $\Delta ((u,v)_w)$ of an interval therein is homotopy equivalent to a sphere if $Q_{u,v}$ is a face of $Q_{e,w}$ and is otherwise contractible. This significantly generalizes the known case of the permutahedron. We also show that saturated chains from $u$ to $v$ in such lattices are connected, and in fact highly connected, under moves corresponding to flipping across a 2-face. When $w$ is a Grassmannian permutation, this implies a strengthening of the restriction of Postnikov's move-equivalence theorem to the class of BCFW bridge decomposable plabic graphs.