Journal of Combinatorics

Volume 14 (2023)

Number 2

Products of reflections in smooth Bruhat intervals

Pages: 197 – 211

DOI: https://dx.doi.org/10.4310/JOC.2023.v14.n2.a3

Authors

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

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

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.

Keywords

smooth permutation, Schubert variety, reflection, compatible order

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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