Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs

TL;DR

This paper develops dual families of symmetric Grothendieck polynomials using Schur operators, proving skew Cauchy identities and deriving applications like Pieri rules and Young lattice filtrations in K-theory.

Contribution

It introduces a novel approach to construct dual symmetric Grothendieck polynomials and proves fundamental identities with multiple applications in algebraic combinatorics.

Findings

Proved skew Cauchy identity for symmetric Grothendieck polynomials

Derived skew Pieri rules and dual filtrations of Young's lattice

Provided a new explanation for finite expansion property in Grothendieck polynomial products

Abstract

Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.