Séminaire Lotharingien de Combinatoire, 85B.5 (2021), 12 pp.
Zachary Hamaker, Oliver Pechenik and Anna Weigandt
Gröbner Geometry of Schubert Polynomials Through Ice
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").
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
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