Gröbner geometry of Schubert polynomials through ice

Zachary R Hamaker; Oliver Pechenik; Anna Weigandt

Gröbner geometry of Schubert polynomials through ice

Zachary R Hamaker, Oliver Pechenik, Anna Weigandt

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

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”).

Original language	English (US)
Article number	#5
Journal	Seminaire Lotharingien de Combinatoire
Issue number	85
State	Published - 2021
Externally published	Yes

Bibliographical note

Funding Information:
∗[email protected] †[email protected] Oliver Pechenik was partially supported by a Mathematical Postdoctoral Research Fellowship (#1703696) from the National Science Foundation. ‡[email protected]

Publisher Copyright:
© 2021, Seminaire Lotharingien de Combinatoire. All Rights Reserved.

Keywords

bumpless pipe dream
matrix Schubert variety
Schubert polynomial

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AU - Hamaker, Zachary R

AU - Pechenik, Oliver

AU - Weigandt, Anna

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PY - 2021

Y1 - 2021

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 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”).

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 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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Gröbner geometry of Schubert polynomials through ice

Abstract

Bibliographical note

Keywords

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