Symmetric Fundamental Expansions to Schur Positivity

Austin Roberts

doi:10.46298/dmtcs.6366

Abstract

We consider families of quasisymmetric functions with the property that if a symmetric function f is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of the families studied, we give a combinatorial description of the Schur coefficients of f. We organize six such families into a poset, where functions in higher families in the poset are always positive integer sums of functions in each of the lower families.

Highlights

In this extended abstract, we consider families of quasisymmetric functions with the property that if a symmetric function f is a positive sum of functions in one of these families, f is necessarily a positive sum of Schur functions
In each of the families studied, we give a combinatorial description of the Schur coefficients of f
Each of the families is defined as sums of fundamental quasisymmetric functions over equivalence classes on standard Young tableaux of fixed partition shape λ, SYT(λ)

Summary

Introduction

We consider families of quasisymmetric functions with the property that if a symmetric function f is a positive sum of functions in one of these families, f is necessarily a positive sum of Schur functions. In each of the families studied, we give a combinatorial description of the Schur coefficients of f We organize six such families into a poset, where functions in higher families in the poset are positive integer sums of functions in each of the lower families. Each of the families is defined as sums of fundamental quasisymmetric functions over equivalence classes on standard Young tableaux of fixed partition shape λ, SYT(λ). Similar to the quasisymmetric Schur functions, Proposition 4.4 states that the generators for shifted dual equivalence strictly contain the generators for ≡2 after applying a simple involution. In Proposition 4.8, we further show that the set of row reading words of shifted standard Young tableaux with a fixed shape comprise an equivalence class of ≡2.

Preliminaries

The equivalence relations on tableaux

The equivalence relations on permutations

Extending to the quasisymmetric Schur functions

Extending to shifted dual equivalence