Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions

TL;DR

This paper establishes two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions, generalizing classical rules and confirming a conjecture about their combinatorial classification.

Contribution

It introduces two new Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions, extending classical results and providing a combinatorial classification.

Findings

Both rules generalize the classical Littlewood-Richardson rule.

Confirmed a conjecture on the classification of symmetric skew quasisymmetric Schur functions.

Contained classical Littlewood-Richardson rule as a special case.

Abstract

The classical Littlewood-Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood-Richardson rule involving Yamanouchi words. Furthermore, both our rules contain this classical Littlewood-Richardson rule as a special case. We then apply our rules to combinatorially classify symmetric skew quasisymmetric Schur functions. This answers affirmatively a conjecture of Bessenrodt, Luoto and van Willigenburg.