Gröbner geometry of Schubert polynomials through ice

Zachary Hamaker, Oliver Pechenik, Anna Weigandt

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5 Scopus citations

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 combinatonarics 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 (2021) in the guise of “bumpless pipe dreams”).

Original languageEnglish (US)
Article number108228
JournalAdvances in Mathematics
Volume398
DOIs
StatePublished - Mar 26 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

Keywords

  • Bumpless pipe dream
  • Gröbner bases
  • Matrix Schubert variety
  • Schubert polynomial

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