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September 5, 2025Open Access

Degrees of P-Grothendieck polynomials and regularity of Pfaffian varieties

Key Points

The study presents a new formula for degrees of Grothendieck polynomials, enhancing understanding of their combinatorial aspects.
Key evidence indicates that new findings relate to the regularity of pfaffian varieties, specifically vexillary skew-symmetric matrix Schubert varieties.
An observational analysis showcases the application of shifted tableaux, leading to improved insights on Grassmannian matrix Schubert varieties.
These findings support the ongoing exploration of combinatorial methods in algebraic geometry, contributing valuable knowledge in the field.

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.

Degrees of P-Grothendieck polynomials and regularity of Pfaffian varieties

Key Points

Abstract

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