Stability of products of double Grothendieck polynomials

TL;DR

This paper investigates the stability properties of products of double Grothendieck polynomials, establishing their relation to Schubert polynomials and confirming a finiteness conjecture with new proofs and characterizations.

Contribution

It provides new insights into the stability of double Grothendieck polynomial products, characterizes simple reflections involved, and offers a novel proof of a key finiteness conjecture.

Findings

Products share stability numbers with Schubert polynomials

Characterization of simple reflections in such products

New proof of Lam-Lee-Shimozono's finiteness conjecture

Abstract

We prove that products of double Grothendieck polynomials have the same back- and forward-stability numbers as products of Schubert polynomials, characterize which simple reflections appear in such products, and also give a new proof of a finiteness conjecture of Lam-Lee-Shimozono on products of back-stable Grothendieck polynomials which was first proved by Anderson. To do this, we use the main theorems from our recent work, as well as expansion formulas of Lenart, Fomin-Kirillov, and Lam-Lee-Shimozono.