Multiplicity free Schur, skew Schur, and quasisymmetric Schur functions

TL;DR

This paper classifies all Schur, skew Schur, and quasisymmetric Schur functions that are multiplicity free when expanded in the fundamental quasisymmetric basis, revealing combinatorial structures with distinct descent sets.

Contribution

It provides a comprehensive classification of F-multiplicity free Schur and skew Schur functions, extending to quasisymmetric Schur functions with limited expansion terms.

Findings

Classified all F-multiplicity free Schur functions and skew shapes.

Identified composition shapes with standard tableaux having distinct descent sets.

Provided a classification for quasisymmetric Schur functions that are nearly F-multiplicity free.

Abstract

In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape have distinct descent sets. We conclude by providing such a classification for quasisymmetric Schur function families, giving a classification of Schur functions that are in some sense almost F-multiplicity free.