Gröbner geometry of Schubert polynomials through ice

Zachary R Hamaker; Oliver Pechenik; Anna Weigandt

doi:10.1016/j.aim.2022.108228

Gröbner geometry of Schubert polynomials through ice

Zachary R Hamaker, Oliver Pechenik, Anna Weigandt

Research output: Contribution to journal › Article › peer-review

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 language	English (US)
Article number	108228
Journal	Advances in Mathematics
Volume	398
DOIs	https://doi.org/10.1016/j.aim.2022.108228
State	Published - Mar 26 2022
Externally published	Yes

Bibliographical note

Funding Information:
ZH was partially supported by National Science Foundation grant DMS-2054423 . OP was partially supported by a Mathematical Sciences Postdoctoral Research Fellowship (# 1703696 ) from the National Science Foundation . OP also acknowledges support from NSERC Discovery Grant RGPIN-2021-02391 and Launch Supplement DGECR-2021-00010 . AW was partially supported by Bill Fulton's Oscar Zariski Distinguished Professor Chair funds.

Publisher Copyright:
© 2022 Elsevier Inc.

Keywords

Bumpless pipe dream
Gröbner bases
Matrix Schubert variety
Schubert polynomial

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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{\"o}bner degeneration of matrix Schubert varieties. We consider instead diagonal Gr{\"o}bner 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”).",

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AU - Pechenik, Oliver

AU - Weigandt, Anna

N1 - Funding Information: ZH was partially supported by National Science Foundation grant DMS-2054423 . OP was partially supported by a Mathematical Sciences Postdoctoral Research Fellowship (# 1703696 ) from the National Science Foundation . OP also acknowledges support from NSERC Discovery Grant RGPIN-2021-02391 and Launch Supplement DGECR-2021-00010 . AW was partially supported by Bill Fulton's Oscar Zariski Distinguished Professor Chair funds. Publisher Copyright: © 2022 Elsevier Inc.

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N2 - 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”).

AB - 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”).

KW - Bumpless pipe dream

KW - Gröbner bases

KW - Matrix Schubert variety

KW - Schubert polynomial

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ER -

Gröbner geometry of Schubert polynomials through ice

Abstract

Bibliographical note

Keywords

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