Integrability approach to Feher-Nemethi-Rimanyi-Guo-Sun type identities for factorial Grothendieck polynomials

TL;DR

This paper explores the integrability of identities related to factorial Grothendieck polynomials, providing new proofs and duality formulas through quantum integrability methods and extending results to rectangular Young diagrams.

Contribution

It introduces an integrability-based approach to prove and extend identities for factorial Grothendieck polynomials, including duality formulas and $q$-deformations.

Findings

New proof of Guo-Sun identity using quantum inverse scattering

Derivation of an identity for rectangular Young diagrams

Establishment of a duality formula for factorial Grothendieck polynomials

Abstract

Recently, Guo and Sun derived an identity for factorial Grothendieck polynomials which is a generalization of the one for Schur polynomials by Feh\'er, N\'emethi and Rim\'anyi. We analyze the identity from the point of view of quantum integrability, based on the correspondence between the wavefunctions of a five-vertex model and the Grothendieck polynomials. We give another proof using the quantum inverse scattering method. We also apply the same idea and technique to derive an identity for factorial Grothendieck polynomials for rectangular Young diagrams. Combining with the Guo-Sun identity, we get a duality formula. We also discuss a $q$-deformation of the Guo-Sun identity.