Charge on tableaux and the poset of k-shapes

TL;DR

This paper introduces a charge statistic on k-tableaux, linking it to the poset of k-shapes and providing a combinatorial interpretation for expansion coefficients of Hall-Littlewood polynomials into k-Schur functions.

Contribution

It defines a new charge concept on k-tableaux and proves its compatibility with the weak bijection, connecting tableau charges to paths in the poset of k-shapes.

Findings

Charge of standard tableaux equals sum of path charges in the poset for k=2,...,n

Compatibility of charge with the weak bijection in the standard case

Provides a combinatorial interpretation for Hall-Littlewood polynomial expansions

Abstract

A poset on a certain class of partitions known as k-shapes was recently introduced to provide a combinatorial rule for the expansion of a (k-1)-Schur functions into k-Schur functions at t=1. The main ingredient in this construction was a bijection, which we call the weak bijection, that associates to a k-tableau a pair made out of a (k-1)-tableau and a path in the poset of k-shapes. We define here a concept of charge on k-tableaux (which conjecturally gives a combinatorial interpretation for the expansion coefficients of Hall-Littlewood polynomials into k-Schur functions), and show that it is compatible in the standard case with the weak bijection. In particular, we obtain that the usual charge of a standard tableau of size n is equal to the sum of the charges of its corresponding paths in the poset of k-shapes, for k=2,3...n.