Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

TL;DR

This paper explores the structure of parabolic Tamari lattices derived from the weak order on permutations, using modified arc diagrams to better understand their properties and relationships.

Contribution

It introduces modified arc diagrams to analyze parabolic Tamari lattices, extending the combinatorial models for these quotient lattices of the symmetric group.

Findings

Structural insights into parabolic Tamari lattices

Modified arc diagrams facilitate understanding of lattice properties

Connections between arc diagrams and quotient lattice structures

Abstract

Ordering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.