Zircons and smooth Bruhat intervals in symmetric groups

TL;DR

This paper explores the relationship between zircons and smooth Bruhat intervals in symmetric groups, establishing conditions for rational smoothness and proposing conjectures linking various properties in type A.

Contribution

It proves that duals of Bruhat intervals that are zircons imply rational smoothness and introduces conjectures connecting smoothness, zircons, and isomorphisms in type A.

Findings

Duals of zircons imply rational smoothness.

Conjectures suggest equivalence of properties in type A.

Verification of conjectures up to type A8.

Abstract

In this paper, we prove that if the dual of a Bruhat interval in a Weyl group is a zircon, then that interval is rationally smooth. Investigating when the converse holds, and drawing inspiration from conjectures by Delanoy, leads us to pose two conjectures. If true, they imply that for Bruhat intervals in type $A$, duals of smooth intervals, zircons, and being isomorphic to lower intervals are all equivalent. As a verification, we have checked our conjectures in types $A_n$, $n\leq 8$.