A Proof of Rubey's Lattice Conjecture

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.