A Proof of Rubey's Lattice Conjecture

TL;DR

This work resolves Rubey's lattice conjecture by introducing the move operator ${\mathcal{M}}_{ij}$ and a constructive join algorithm for reduced pipe dreams. It proves that for every permutation $w$, the Rubey poset $( {\textnormal{RPD}}(w),<)$ is a lattice, with meets obtained via transpose and joins computed through a recursive procedure based on principal disagreements. A comparability criterion is provided using pipe dream tableaux ${\mathbf{T}}(D)$, enabling efficient comparison and enabling bounds on the total number of reduced pipe dreams and maximal chain length. The results connect to Schubert polynomials and offer a concrete, tableau-driven perspective on Rubey lattices, with several open directions for explicit tableau characterizations and sampling on Rubey lattices.

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.

Figures (13)

• Figure 1: A reduced pipe dream $D$ for $w=[3,1,4,6,5,2]$.
• Figure 2: The diagram of $w=43152$ is the set of outlined cells above, so $\textbf{D}(w)=\{(1,1), (1,2), (1,3), (2,1), (2,2), (4,2) \}$ using matrix coordinates.
• Figure 3: Chute moves on pipe dreams move one cross down and to the left, preserving the permutation.
• Figure 4: Ladder moves on pipe dreams move one cross up and to the right, preserving the permutation.
• Figure 5: A pipe dream in which the cross tile at $(3,1)$ is ladder movable by swapping it with the bump tile $(1,5)$ is shown in the simplified form and with the wire tiles drawn.

Theorems & Definitions (92)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Example 2.4
• Definition 3.1
• Definition 3.2
• Example 3.3