The $\nu$-Tamari lattice via $\nu$-trees, $\nu$-bracket vectors, and subword complexes

TL;DR

This paper offers new combinatorial and geometric interpretations of the $ u$-Tamari lattice, connecting it with $ u$-trees, bracket vectors, subword complexes, and polytopal subdivisions, advancing understanding of its structure and properties.

Contribution

It introduces a rotation lattice of $ u$-trees, relates the $ u$-Tamari lattice to subword complexes, and explores its generalization to multi $ u$-Tamari complexes.

Findings

$ u$-Tamari lattice is a rotation lattice of $ u$-trees

It is isomorphic to the increasing-flip poset of a subword complex

Provides insight into geometric realizability of multi $ u$-Tamari complexes

Abstract

We give new interpretations of the $\nu$-Tamari lattice of Pr\'eville-Ratelle and Viennot. First, we describe it as a rotation lattice of $\nu$-trees, which uncovers the relation with known combinatorial objects such as tree-like tableaux and north-east fillings. Then, using a formulation in terms of bracket vectors of $\nu$-trees and componentwise order, we provide a simple description of the lattice property. We also show that the $\nu$-Tamari lattice is isomorphic to the increasing-flip poset of a suitably chosen subword complex, and settle a special case of Rubey's lattice conjecture concerning the poset of pipe dreams defined by chute moves. Finally, this point of view generalizes to multi $\nu$-Tamari complexes, and gives (conjectural) insight on their geometric realizability via polytopal subdivisions of multiassociahedra.