Martin Rubey: Invariance of rowmotion for variants of the Tamari lattice

Invariance of rowmotion for variants of the Tamari lattice

Martin Rubey

Joint work in progress with Ben Adenbaum, Emily Barnard, Cesar Ceballos, Clément Chenevière, Colin Defant, Sam Hopkins, Matthias Müller & Jessica Striker.

Abstract: Two classical lattices on Dyck paths are the Stanley lattice, where the covering relation turns a valley into a peak, and Tamari lattice, where the covering relation exchanges a down step with the prime Dyck path following it. The alt-$\nu$-Tamari lattices, depending on a north-east path $\nu$ and a vector of non-negative integers $\delta$, are a common generalisation of these two.

Alt-$\nu$-Tamari lattices are semidistributive. This implies that they carry a natural cyclic action known as rowmotion, which generalises classical rowmotion on the distributive lattice of order ideals of a poset.

We show that, surprisingly, the rowmotion operator has the same orbit structure for all alt-$\nu$-Tamari lattices with given path $\nu$. In fact, we conjecture that this invariance of rowmotion extends to the even more general cross Tamari lattices.