Zircons and smooth Bruhat intervals in symmetric groups

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 $A_n$, $n\leq 8$.