The expansion of Hall-Littlewood functions in the dual Grothendieck polynomial basis

Jason Bandlow,Jennifer Morse

doi:10.46298/dmtcs.2860

Abstract

A combinatorial expansion of the Hall-Littlewood functions into the Schur basis of symmetric functions was first given by Lascoux and Schützenberger, with their discovery of the charge statistic. A combinatorial expansion of stable Grassmannian Grothendieck polynomials into monomials was first given by Buch, using set-valued tableaux. The dual basis of the stable Grothendieck polynomials was given a combinatorial expansion into monomials by Lam and Pylyavskyy using reverse plane partitions. We generalize charge to set-valued tableaux and use all of these combinatorial ideas to give a nice expansion of Hall-Littlewood polynomials into the dual Grothendieck basis. \par En associant une charge à un tableau, une formule combinatoire donnant le développement des polynômes de Hall-Littlewood en termes des fonctions de Schur a été obtenue par Lascoux et Schützenberger. Une formule combinatoire donnant le développement des polynômes de Grothendieck Grassmanniens stables en termes des fonctions monomiales a quant à elle été obtenue par Buch à l'aide de tableaux à valeurs sur des ensembles. Finalement, une formule faisant intervenir des partitions planaires inverses a été obtenue par Lam et Pylyavskyy pour donner le développement de la base duale aux polynômes de Grothendieck stables en termes de monômes. Nous généralisons le concept de charge aux tableaux à valeurs sur des ensembles et, en nous servant de toutes ces notions combinatoires, nous obtenons une formule élégante donnant le développement des polynômes de Hall-Littlewood en termes de la base de Grothendieck duale.

Highlights

The Hall-Littlewood functions are symmetric functions with a wealth of applications
The stable Grothendieck polynomials introduced by Fomin and Kirillov [FK94] are the stable limit of these as the number of variables approaches infinity
These functions, written Gλ, are non-homogeneous symmetric functions, which cannot be written as a finite sum of Schur functions

Summary

Introduction

The Hall-Littlewood functions are symmetric functions with a wealth of applications. In various forms, they interpolate between the complete homogeneous and Schur basis of symmetric functions, provide a polynomial realization of the Hall algebra, and have several interpretations as characters of representations. Lascoux and Schutzenberger gave an expansion of the Hall-Littlewood functions into Schur functions, in terms of a statistic on tableaux called charge [LS78]. The stable Grothendieck polynomials introduced by Fomin and Kirillov [FK94] are the stable limit of these as the number of variables approaches infinity These functions, written Gλ, are non-homogeneous symmetric functions, which cannot be written as a finite sum of Schur functions. Buch gave an expansion of the stable Grothendieck polynomials into monomial symmetric functions by introducing set-valued tableaux [Buc02]. Lam and Pylyavskyy studied the dual basis to the stable Grothendieck polynomials under the Hall inner product [LP07] They expanded these into monomials using a special evaluation of reverse plane partitions. We find it remarkable that such a nice formula exists, as we are unaware of any direct connection between the HallLittlewood functions and K-theory

Symmetric function basics

Hall-Littlewood symmetric functions

Grothendieck polynomials

Main result

Definition of the involution