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

Research output: Contribution to journalResearch articleContributedpeer-review

Contributors

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.

Details

Original languageEnglish
Pages (from-to)307-344
Number of pages38
JournalAnnals of combinatorics
Volume25
Issue number2
Publication statusPublished - Jun 2021
Peer-reviewedYes

External IDs

Scopus 85104236065

Keywords

Keywords

  • Noncrossing arc diagrams, Tamari lattices, Congruence-uniform lattices, Trim lattices, Core label order, Parabolic quotients, Symmetric group

Library keywords