The $\nu$-Tamari lattice via $\nu$-trees, $\nu$-bracket vectors, and subword complexes
Abstract
We give new interpretations of the -Tamari lattice of Pr\'eville-Ratelle and Viennot. First, we describe it as a rotation lattice of -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 -trees and componentwise order, we provide a simple description of the lattice property. We also show that the -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 -Tamari complexes, and gives (conjectural) insight on their geometric realizability via polytopal subdivisions of multiassociahedra.
Keywords
Cite
@article{arxiv.1805.03566,
title = {The $\nu$-Tamari lattice via $\nu$-trees, $\nu$-bracket vectors, and subword complexes},
author = {Cesar Ceballos and Arnau Padrol and Camilo Sarmiento},
journal= {arXiv preprint arXiv:1805.03566},
year = {2019}
}
Comments
26 pages, 20+ figures. v2: added correspondence between binary trees and v-trees (Lemma 2.4); added Section 5.3 about connection to Woo's bijection using Edeleman-Greene correspondence; rephrased Theorem 5.5; other minor changes