A Proof of Rubey's Lattice Conjecture

TL;DR

This paper proves Rubey's conjecture that the poset of reduced pipe dreams under generalized moves forms a lattice, introducing new move operations and recursive formulas to establish lattice properties.

Contribution

We prove Rubey's lattice conjecture by defining new move operations and deriving recursive formulas for joins and meets in the poset.

Findings

Rubey's poset is a lattice.

Introduced move operation $\\mathcal{M}_{ij}$ for poset analysis.

Provided criteria for comparability in Rubey's poset.

Abstract

In 2011, Rubey generalized chute and ladder moves on the set of reduced pipe dreams for a permutation $w$ and conjectured that the induced poset on reduced pipe dreams is a lattice. In this paper, we prove this conjecture. Our key tool is a new type of move operation $\mathcal{M}_{ij}$, defined as a composite of certain general ladder moves in Rubey's poset. We show that joins and meets exist in Rubey's poset by proving simple recursive formulas in terms of $\mathcal{M}_{ij}$ operations. In addition, we give an explicit criterion to determine if two elements of Rubey's poset are comparable.