Degrees of $P$-Grothendieck polynomials and regularity of Pfaffian varieties

Pechenik, Oliver; St.Denis, Matthew

doi:10.5802/alco.431

Pechenik, Oliver ¹; St.Denis, Matthew ¹

¹ University of Waterloo Dept. of Combinatorics & Optimization Waterloo, ON, N2L 3G1 (Canada)

Algebraic Combinatorics, Volume 8 (2025) no. 4, pp. 897-923.

Abstract

We prove a formula for the degrees of Ikeda and Naruse’s $P$-Grothendieck polynomials using combinatorics of shifted tableaux. We show this formula can be used in conjunction with results of Hamaker, Marberg, and Pawlowski to obtain an upper bound on the Castelnuovo–Mumford regularity of certain Pfaffian varieties known as vexillary skew-symmetric matrix Schubert varieties. Similar combinatorics additionally yields a new formula for the degree of Grassmannian Grothendieck polynomials and the regularity of Grassmannian matrix Schubert varieties, complementing a 2021 formula of Rajchgot, Ren, Robichaux, St. Dizier, and Weigandt.

Received: 2024-08-30
Revised: 2025-01-31
Accepted: 2025-03-25
Published online: 2025-09-01

DOI: 10.5802/alco.431

Classification: 05E05, 05E40, 13C40, 14M15
Keywords: Pfaffian variety, $P$-Grothendieck polynomial, regularity, skew-symmetric matrix Schubert variety

Author's affiliations:

Pechenik, Oliver ¹; St.Denis, Matthew ¹

¹ University of Waterloo Dept. of Combinatorics & Optimization Waterloo, ON, N2L 3G1 (Canada)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Pechenik, Oliver and St.Denis, Matthew},
     title = {Degrees of $P${-Grothendieck} polynomials and regularity of {Pfaffian} varieties},
     journal = {Algebraic Combinatorics},
     pages = {897--923},
     publisher = {The Combinatorics Consortium},
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Pechenik, Oliver; St.Denis, Matthew. Degrees of $P$-Grothendieck polynomials and regularity of Pfaffian varieties. Algebraic Combinatorics, Volume 8 (2025) no. 4, pp. 897-923. doi : 10.5802/alco.431. https://alco.centre-mersenne.org/articles/10.5802/alco.431/

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