Poset topology, moves, and Bruhat interval polytope lattices

2024/10/10 by Gaetz, Christian, Hersh, Patricia
#05E45 #06A07 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.08076

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.

Related

• Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology
• Bruhat Interval Polytopes
• Chute Move Posets are Lattices
• A positive Grassmannian analogue of the permutohedron
• Signed Poset Polytopes