A unifying framework for the $\nu$-Tamari lattice and principal order ideals in Young's lattice

TL;DR

This paper introduces a unifying geometric framework connecting the $ u$-Tamari lattice and principal order ideals in Young's lattice through flow polytope triangulations, revealing new combinatorial and geometric insights.

Contribution

It provides a new geometric realization of the $ u$-Tamari complex and extends the understanding of flow polytope subdivisions, unifying previous results in the field.

Findings

The $h^*$-vector of the $ u$-caracol flow polytope equals the $ u$-Narayana numbers.

A new geometric realization of the $ u$-Tamari complex is constructed.

The work generalizes and unifies dual structures of polytope subdivisions studied by Pitman and Stanley.

Abstract

We present a unifying framework in which both the $\nu$-Tamari lattice, introduced by Pr\'eville-Ratelle and Viennot, and principal order ideals in Young's lattice indexed by lattice paths $\nu$, are realized as the dual graphs of two combinatorially striking triangulations of a family of flow polytopes which we call the $\nu$-caracol flow polytopes. The first triangulation gives a new geometric realization of the $\nu$-Tamari complex introduced by Ceballos, Padrol and Sarmiento. We use the second triangulation to show that the $h^*$-vector of the $\nu$-caracol flow polytope is given by the $\nu$-Narayana numbers, extending a result of M\'esz\'aros when $\nu$ is a staircase lattice path. Our work generalizes and unifies results on the dual structure of two subdivisions of a polytope studied by Pitman and Stanley.