A bijective proof of the Cauchy identity for Grothendieck polynomials

TL;DR

This paper provides a bijective proof of the Cauchy identity for Grothendieck polynomials using algorithms on pairs of set-valued tableaux and reverse plane partitions, offering a combinatorial perspective.

Contribution

It introduces algorithms that establish a bijective proof of the finite sum Cauchy identity for Grothendieck and dual Grothendieck polynomials, a novel combinatorial approach.

Findings

Bijection between set-valued tableaux and reverse plane partitions

Proof of the Cauchy identity for Grothendieck polynomials

New algorithms for combinatorial objects

Abstract

We consider pairs of a set-valued column-strict tableau and a reverse plane partition of the same shape. We introduce algortithms for them, which implies a bijective proof for the finite sum Cauchy identity for Grothendieck polynomials and dual Grothendieck polynomials.