Poset topology, moves, and Bruhat interval polytope lattices

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.

💡 Research Summary

The paper investigates the combinatorial and topological structure of the poset (P_w) obtained from the 1‑skeleton of a Bruhat interval polytope (Q_{e,w}) after orienting its edges by a generic cost vector. The authors focus on two central questions: (1) what is the homotopy type of the order complex of an open interval ((u,v)_w) in (P_w), and (2) how are the maximal chains (i.e., facets of that order complex) connected via elementary “flips’’ across 2‑dimensional faces.

Main results.
Theorem 1.1 states that for any open interval ((u,v)w) the order complex (\Delta((u,v)w)) is homotopy equivalent to a sphere of dimension (|A_w(u,v)|-2) precisely when the Bruhat interval polytope (Q{u,v}) is a face of (Q{e,w}); otherwise the complex is contractible. Consequently the Möbius function of (P_w) takes the values ((-1)^{|A_w(u,v)|}) or (0) accordingly. The proof hinges on a “face non‑revisiting’’ property (any directed path staying inside a face once it enters) and on showing that distinct subsets of atoms have distinct joins (Proposition 3.2). This yields a surjective order‑preserving map from (