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 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.Lemma 46. Let c n be the cover corresponding to letter n in the k-shape tableau T . Then the charge and cocharge of the letter n satisfy the relation ch(n) = n − coch(n) − |c n | (9.9)Proof. We proceed by induction. The case n = 1 is easily seen to hold. We need to establish that ch(n + 1) = n + 1 − coch(n + 1) − |c n+1 | (9.10)Using the induction hypothesis, it suffices to prove that ch(n + 1) − ch(n) + coch(n + 1) − coch(n) = |c n | − |c n+1 | + 1 (9.11)We first consider the case where n ↑ + and n ↓ + are above n ↑ and n ↓ . We can visualize this case with a diagram. The symbol ⋆ denotes the rows that are k-connected with the row of n ↑ + , while ⋄ denote the rows above the letters n.

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

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