The extra slow Tamari lattice

TL;DR

The paper introduces the extra slow Tamari lattices, a new family of lattices based on faithfully balanced tableaux linked to type A quiver representation theory, extending classical Tamari lattices.

Contribution

It defines and characterizes the extra slow Tamari lattices, proving their lattice properties and exploring their structural, enumerative, and combinatorial aspects.

Findings

They are lattices, semidistributive, trim, polygonal, and congruence uniform.

Explicit descriptions of meets, joins, and join-irreducible elements.

Enumerative results for the new lattices and their related structures.

Abstract

We introduce the extra slow Tamari lattices, a new family of lattices defined on faithfully balanced tableaux. These tableaux arise naturally from the representation theory of type \( A \) quivers, and our construction extends the classical Tamari lattice and the slow Tamari lattice. We explicitly describe meets and joins in the extra slow Tamari lattices, and then prove that they are lattices. We then show that they are semidistributive, trim, polygonal, and congruence uniform. Their join-irreducible elements are described in terms of a three-color analogue of the positive roots of type \( A \), which leads to descriptions of their spines and congruence lattices. We also obtain several enumerative results for the extra slow Tamari lattices and their associated structures. Finally, we derive new structural and enumerative results for the slow Tamari lattices.