Interval structure of the Pieri formula for Grothendieck polynomials

TL;DR

This paper provides a combinatorial interpretation of the Pieri formula for double Grothendieck polynomials using Bruhat order intervals, offering a new proof and structural insight into the permutations involved.

Contribution

It introduces a new combinatorial interpretation of the Pieri formula for Grothendieck polynomials via Bruhat order intervals, connecting algebraic and order-theoretic perspectives.

Findings

Permutations in the Pieri formula form an interval in Bruhat order

A direct proof of Lenart and Postnikov's result is provided

The interpretation links Grothendieck polynomials with 0-Hecke algebra generators

Abstract

We give a combinatorial interpretation of a Pieri formula for double Grothendieck polynomials in terms of an interval of the Bruhat order. Another description had been given by Lenart and Postnikov in terms of chain enumerations. We use Lascoux's interpretation of a product of Grothendieck polynomials as a product of two kinds of generators of the 0-Hecke algebra, or sorting operators. In this way we obtain a direct proof of the result of Lenart and Postnikov and then prove that the set of permutations occuring in the result is actually an interval of the Bruhat order.