Combinatorial relations on skew Schur and skew stable Grothendieck polynomials

TL;DR

This paper introduces a new combinatorial expansion of stable Grothendieck polynomials for skew shapes in terms of skew Schur functions, unifying previous formulas and providing a two-way expansion between these functions.

Contribution

It presents a novel row insertion algorithm for set-valued tableaux and generalizes multiple existing formulas relating skew Schur and Grothendieck polynomials.

Findings

Unified combinatorial expansion for skew Grothendieck polynomials

New row insertion algorithm for set-valued tableaux

Two-way expansion between skew Schur and Grothendieck functions

Abstract

We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in earlier joint work with L\'opez Mart\'in and Teixidor i Bigas concerning Brill-Noether curves, and it generalizes a 2000 formula of Lenart and a recent result of Reiner-Tenner-Yong to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms of skew Grothendieck polynomials.