Products of reflections in smooth Bruhat intervals

TL;DR

This paper characterizes smooth permutations in the symmetric group by showing that products of reflections below a smooth element in Bruhat order form saturated chains, providing a new criterion for smoothness.

Contribution

It extends previous results by proving that such products uniquely determine saturated chains and characterizes smooth elements through this property.

Findings

Products of reflections below smooth elements form saturated chains in Bruhat order

This chain property characterizes smooth permutations in the symmetric group

Strengthens previous results by Gilboa and Lapid

Abstract

A permutation is called smooth if the corresponding Schubert variety is smooth. Gilboa and Lapid prove that in the symmetric group, multiplying the reflections below a smooth element $w$ in Bruhat order in a compatible order yields back the element $w$. We strengthen this result by showing that such a product in fact determines a saturated chain $e \to w$ in Bruhat order, and that this property characterizes smooth elements.