Charge on tableaux and the poset of k-shapes

A poset on a certain class of partitions known as k-shapes was introduced in [7] to provide a combinatorial rule for the expansion of a k − 1 -Schur function 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 .