Combinatorial relations on skew Schur and skew stable Grothendieck polynomials

Melody Chan,Nathan Pflueger

doi:10.5802/alco.144

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 López Martín 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.

Highlights

Let σ be a skew Young diagram (Definition 2.1)
The main result of this paper is a new formula for the skew stable Grothendieck polynomial Gσ of Lascoux–Schützenberger and Fomin–Kirillov [11, 13] as a linear combination of skew Schur functions sλ on related shapes λ
As was demonstrated by Buch, the coefficients of Gσ have a combinatorial interpretation in terms of set-valued tableaux [5], and our original motivation for this paper came from a recent geometric result, proved in a companion paper [8], identifying Euler characteristics of Brill–Noether varieties up to sign as counts of set-valued standard tableaux

Summary

Introduction

The main result of this paper is a new formula for the skew stable Grothendieck polynomial Gσ of Lascoux–Schützenberger and Fomin–Kirillov [11, 13] as a linear combination of skew Schur functions sλ on related shapes λ. It is natural to ask for a linear expansion of Gσ in terms of other symmetric functions, the basis of Schur functions. Such an expansion was obtained by Fomin–Greene, who obtained such expansions for a wide class of symmetric functions including stable Grothendieck polynomials associated to arbitrary permutations [10]. Our main theorem expresses Gσ instead as a linear combination of skew Schur functions sλ. Grothendieck polynomials, insertion algorithms, set-valued tableaux, Brill–Noether theory

Preliminaries

Row insertion for skew set-valued tableaux

Findings

An inverse formula