Degrees of P-Grothendieck polynomials and regularity of Pfaffian varieties

TL;DR

This paper derives formulas for degrees of P-Grothendieck polynomials and uses combinatorics to establish bounds on the regularity of Pfaffian varieties, advancing understanding in algebraic geometry and combinatorics.

Contribution

It introduces new combinatorial formulas for P-Grothendieck polynomials and applies them to bound the regularity of Pfaffian and Grassmannian Schubert varieties.

Findings

Derived a formula for degrees of P-Grothendieck polynomials.

Established an upper bound on Castelnuovo-Mumford regularity of Pfaffian varieties.

Provided a new formula for degrees and regularity of Grassmannian Grothendieck polynomials.

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.