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Gr\"obner geometry of Schubert polynomials through ice

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arxiv 2003.13719 v1 pith:A2N5GHHN submitted 2020-03-30 math.CO math.ACmath.AG

classification math.COmath.ACmath.AG
keywords schubertobnerdegenerationsdiagonalknutsonmatrixmillerpipe
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abstract

The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gr\"obner degeneration of matrix Schubert varieties. We consider instead diagonal Gr\"obner 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'').

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  1. Zero-one dual characters of flagged Weyl modules

    math.CO 2024-11 conditional novelty 8.0 of 10

    The dual flagged Weyl character is zero-one if and only if the defining diagram is multiplicity-free, namely it avoids twelve listed subconfigurations up to column swap.

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