Gröbner geometry of Schubert polynomials through ice

Zachary R Hamaker, Oliver Pechenik, Anna Weigandt

Research output: Contribution to journalArticlepeer-review

Abstract

The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of “vexillary” matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of Lascoux (2002) in terms of the 6-vertex ice model (recently rediscovered by Lam, Lee, and Shimozono (2018) in the guise of “bumpless pipe dreams”).

Original languageEnglish (US)
Article number#5
JournalSeminaire Lotharingien de Combinatoire
Issue number85
StatePublished - 2021
Externally publishedYes

Bibliographical note

Funding Information:
∗zhamaker@ufl.edu †oliver.pechenik@uwaterloo.ca Oliver Pechenik was partially supported by a Mathematical Postdoctoral Research Fellowship (#1703696) from the National Science Foundation. ‡weigandt@umich.edu

Publisher Copyright:
© 2021, Seminaire Lotharingien de Combinatoire. All Rights Reserved.

Keywords

  • bumpless pipe dream
  • matrix Schubert variety
  • Schubert polynomial

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