Journal of Combinatorics

Volume 14 (2023)

Number 2

Products of reflections in smooth Bruhat intervals

Pages: 197 – 211



Christian Gaetz (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Ram K. Goel (Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)


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.


smooth permutation, Schubert variety, reflection, compatible order

C. Gaetz was supported by an NSF Mathematical Science Postdoctoral Research Fellowship under grant no. DMS-2103121.

Received 3 November 2021

Published 28 December 2022