Topology of posets with special partial matchings

TL;DR

This paper introduces pircons, a class of posets with special partial matchings, and proves their intervals are PL balls or spheres, extending known results and confirming several conjectures in the field.

Contribution

It defines pircons as a generalization of zircons and proves their intervals are topologically well-behaved, extending previous work on special matchings and poset topology.

Findings

Intervals in pircons are PL balls or spheres

Bruhat orders on certain sets are examples of pircons

The results extend and confirm several existing conjectures

Abstract

Special partial matchings (SPMs) are a generalisation of Brenti's special matchings. Let a \emph{pircon} be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti's zircons. We prove that every open interval in a pircon is a PL ball or a PL sphere. It is then demonstrated that Bruhat orders on certain twisted identities and quasiparabolic $W$-sets constitute pircons. Together, these results extend a result of Can, Cherniavsky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.