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arXiv · 2410.08076

Poset topology, moves, and Bruhat interval polytope lattices

Christian Gaetz·Patricia Hersh
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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.

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Christian Gaetz, Patricia Hersh. 2024-10-10. Poset topology, moves, and Bruhat interval polytope lattices. https://doi.org/10.1112/blms.70309

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