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Poset topology, moves, and Bruhat interval polytope lattices

Christian Gaetz, Patricia Hersh

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 $Δ((u,v)_w)$ 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.

Poset topology, moves, and Bruhat interval polytope lattices

Abstract

We study the poset topology of lattices arising from orientations of 1-skeleta of directionally simple polytopes, with Bruhat interval polytopes as our main example. We show that the order complex of an interval therein is homotopy equivalent to a sphere if is a face of and is otherwise contractible. This significantly generalizes the known case of the permutahedron. We also show that saturated chains from to in such lattices are connected, and in fact highly connected, under moves corresponding to flipping across a 2-face. When 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.

Paper Structure

This paper contains 7 sections, 22 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bruhat interval polytopes
  4. Poset topology
  5. Topology of simple 1-skeleton lattices
  6. Poset topology of Bruhat interval polytopes
  7. Higher connectivity of flip graphs

Key Result

Theorem 1.1

The order complex $\Delta((u,v)_w)$ of any open interval in $P_w$ is homotopy equivalent to a sphere $\mathbb{S}^{|A_w(u,v)|-2}$ if $Q_{u,v}$ is a face of $Q_w$, and is otherwise contractible. Thus the Möbius function takes values $\mu_{P_w}(u,v) = (-1)^{|A_w(u,v)|}$ or $\mu_{P_w}(u,v)=0$ accordingl

Figures (3)

  • Figure 1: The Bruhat interval polytope $Q_{3412}$.
  • Figure 2: An example showing some of the chains in the path $\gamma_1 * \gamma_2$ constructed in \ref{['lem:path-exists']}. The first flip is across the face $F"$.
  • Figure 3: An example of a sequence $(\epsilon_0,F_1,\epsilon_1,\dots ,F_r,\epsilon_r)$ from \ref{['lem:unique-sequence']} with $r=2$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1: Kodama--Williams KW
  • Definition 2.2
  • Theorem 2.3: Tsukerman-Williams
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 37 more