On the Degree of Grothendieck Polynomials

A beautiful degree formula for Grothendieck polynomials was recently given by Pechenik, Speyer, and Weigandt (2021). We provide an alternative proof of their degree formula, utilizing the climbing chain model for Grothendieck polynomials introduced by Lenart, Robinson, and Sottile (2006). Moreover for any term order satisfying x 1 <x 2 <⋯<x n , we present the leading monomial of each homogeneous component of the Grothendieck polynomial 𝔊 w (x), confirming a conjecture of Hafner (2022). We conclude with a conjecture for the leading monomial of each homogeneous component of 𝔊 w (x) in any term order satisfying x 1 >x 2 >⋯>x n .