Chute Move Posets are Lattices

TL;DR

This paper proves that the set of reduced pipe dreams for any permutation, ordered by chute moves, forms a lattice, confirming a conjecture and revealing its structure as a semidistributive polygonal lattice.

Contribution

It establishes that chute move posets of reduced pipe dreams are lattices and provides a global description via Lehmer tableaux, confirming Rubey's conjecture.

Findings

$ ext{PD}(w)$ is a lattice for all permutations $w$.

$ ext{PD}(w)$ is isomorphic to Lehmer tableaux poset.

$ ext{PD}(w)$ is a semidistributive polygonal lattice with diamonds and pentagons.

Abstract

For each permutation $w$, we consider the set $\mathrm{PD}(w)$ of reduced pipe dreams for $w$, partially ordered so that cover relations correspond to (generalized) chute moves. Settling a conjecture of Rubey from 2012, we prove that $\mathrm{PD}(w)$ is a lattice. To establish this result, we provide a global description of the partial order on $\mathrm{PD}(w)$ by showing that $\mathrm{PD}(w)$ is isomorphic to a poset consisting of objects called Lehmer tableaux. In addition, we prove that $\mathrm{PD}(w)$ is a semidistributive polygonal lattice whose polygons are all diamonds or pentagons.