A proof of Rubey's lattice conjecture

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 recent work, we prove this conjecture. Our key tool is a new type of move operation \$\mathcal{M}_{ij}\$, defined as a composite of certain generalized ladder moves and later simplified in terms of swaps on a partition shape. 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 the Rubey lattice are comparable using an injective map from reduced pipe dreams to tableaux on the diagram of a permutation. The pipe dream tableaux construction also gives an exact formula for the maximal length of any chain in the Rubey lattice and bounds on the number of reduced pipe dreams for \$w\$, or equivalently the number of terms in the corresponding Schubert polynomial.

This talk is based on joint work with Sara Billey and Connor McCausland. An independent proof of Rubey's conjecture was simultaneously given by Ilani Axelrod-Freed, Colin Defant, Hanna Mularczyk, Son Nguyen, and Katherine Tung.