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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