Vexillary Grothendieck Polynomials via Bumpless Pipe Dreams

TL;DR

This paper provides a new proof for a formula related to vexillary Grothendieck polynomials, characterizes contributing bumpless pipe dreams, and explores their support, connecting existing formulas and addressing conjectures.

Contribution

It offers a novel proof for a vexillary case of the Grothendieck polynomial degree formula and characterizes contributing bumpless pipe dreams, linking different formulas and supporting conjectures.

Findings

Characterization of bumpless pipe dreams contributing to maximal degree terms.

New connections between existing formulas for vexillary Grothendieck polynomials.

Results supporting conjectures on the support of these polynomials.

Abstract

Recent work of Pechenik, Speyer, and Weigandt proved a formula for the degree of any Grothendieck polynomial. A distinct formula for the degree of vexillary Grothendieck polynomials was proven by Rajchgot, Robichaux, and Weigandt. We give a new proof of Pechenik, Speyer, and Weigandt's formula in the special case of vexillary permutations and characterize the set of bumpless pipe dreams which contribute maximal degree terms to the Grothendieck polynomial in this case. Furthermore, we use this characterization to draw connections between the Pechenik-Speyer-Weigandt and Rajchgot-Robichaux-Weigandt formulas. We also use bumpless pipe dreams to prove new results about the support of vexillary Grothendieck polynomials, addressing special cases of conjectures of M\'esz\'aros, Setiabrata, and St. Dizier.