Charge on tableaux and the poset of k-shapes

Abstract

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.

Positivity of affine charge

“…where T is a k-tableau of weight λ. As evidence to support their definition of k-charge, [LP14] prove that the k-charge for a standard k-tableau is compatible with the weak bijection introduced in [LLMS12].…”

“…A massive effort towards the generic t case was put forth by Lapointe and Pinto [LP14]. A key focus in their work is the introduction of a statistic on the set of objects (k-tableaux) whose enumeration is K (k) λµ (1, 1).…”

k-Schur Functions and Affine Schubert Calculus

“…This, in turn, would follow by showing that there is a compatibility between k-charge and the weak bijection introduced in [82]. A partial solution has been given in [99], where the compatibility is shown for standard ktableaux.…”

“…We give here two distinct formulations defined directly on k-tableaux, discovered by Lapointe-Pinto and Morse [99,36]. There are other formulations including one on α-factorizations, one on an object called affine Bruhat countertableau [35,36], and in relation with the energy function on Kirillov-Reshetikhin crystals [126].…”

“…For generic t, the same paper gives a conjecture formula Conjecture 7.8 and discusses the additional properties needed for the result to hold in general. Some progress has been made in this direction in [99].…”