Positivity of affine charge

TL;DR

This paper proves the positivity of affine (co)charge on k-tableaux and confirms a conjecture linking it to Lapointe-Pinto's statistic, advancing understanding of k-Schur functions and their combinatorial properties.

Contribution

It establishes the equivalence of Morse's affine (co)charge with Lapointe-Pinto's statistic and proves positivity for semi-standard tableaux.

Findings

Affine (co)charge is positive for semi-standard tableaux.

Morse's affine (co)charge matches Lapointe-Pinto's statistic.

Positivity of k-(co)charge supports conjectures on k-Schur functions.

Abstract

The branching of (k-1)-Schur functions into k-Schur functions was given by Lapointe, Lam, Morse and Shimozono as chains in a poset on k-shapes. The k-Schur functions are the parameterless case of a more general family of symmetric functions over Q(t), conjectured to satisfy a k-branching formula given by weights on the k-shape poset. A concept of a (co)charge on a k-tableau was defined by Lapointe and Pinto. Although it is not manifestly positive, they prove it is compatible with the k-shape poset for standard k-tableau and the positivity follows. Morse introduced a manifestly positive notion of affine (co)charge on k-tableaux and conjectured that it matches the statistic of Lapointe-Pinto. Here we prove her conjecture and the positivity of k-(co)charge for semi-standard tableaux follows.