Journal of Combinatorics

Volume 13 (2022)

Number 4

Proper permutations, Schubert geometry, and randomness

Pages: 561 – 574

DOI: https://dx.doi.org/10.4310/JOC.2022.v13.n4.a6

Authors

David Brewster (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Il., U.S.A.)

Reuven Hodges (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Il., U.S.A.; and Department of Mathematics, University of California at San Diego, La Jolla, Calif., U.S.A.)

Alexander Yong (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Il., U.S.A.)

Abstract

We define and study proper permutations. Properness is a geometrically natural necessary criterion for a Schubert variety to be Levi-spherical. We prove the probability that a random permutation is proper goes to zero in the limit.

Keywords

Schubert varieties, spherical varieties, proper permutations

2010 Mathematics Subject Classification

Primary 14M15. Secondary 20P05.

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This project was completed as part of the ICLUE (Illinois Combinatorics Lab for Undergraduate Experience) program, which was funded by the NSF RTG grant DMS 1937241.

Reuven Hodges was funded by an AMS Simons Travel grant.

Alexander Yong was funded by a Simons Collaboration grant, and by UIUC’s Center for Advanced Study.

Received 10 September 2021

Accepted 18 September 2021

Published 18 August 2022