Tableau atoms and a new Macdonald positivity conjecture

Abstract

Let Λ be the space of symmetric functions and V k be the subspace spanned by the modified Schur functions {S λ [X/(1 − t)]} λ1≤k . We introduce a new family of symmetric polynomials, {A, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials A (k) λ [X; t] form a basis for V k and that the Macdonald polynomials indexed by partitions whose first part is not larger than k expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but would substantially refine it. Our construction of the A (k) λ [X; t] relies on the use of tableaux combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the A (k) λ [X; t] seem to play the same role for V k as the Schur functions do for Λ. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri and Littlewood-Richardson type coefficients.

How many alphabets can a Schur function accommodate?

“…Having discarded determinants, symmetry, commutativity does not suffice to cover all the generalizations of Schur functions, for example the Macdonald polynomials or the k-Schur functions [14].…”

Alcove random walks, k-Schur functions and the minimal boundary of the k-bounded partition poset

“…, h k ]. The k-Schur functions were introduced by Lascoux, Lapointe and Morse [11] as a basis of Λ (k) . It was established by Lam [7] that their corresponding constant structures (called k-Littlewood-Richardson coefficients) are nonnegative.…”

$k$-Schur functions and affine Schubert calculus

“…In one such study [92], Lapointe, Lascoux, and Morse found computational evidence for a family of new bases for subspaces Λ t k in a filtration Λ t…”

“…They were first defined as a sum of the usual Schur functions over a combinatorially defined collection of tableaux known as a k-atom. Lapointe, Lascoux, and Morse [92] conjectured that the Macdonald symmetric functions expand positively in terms of k-Schur functions. An obvious difficulty with this approach is a missing algebraic connection that could be used to connect Macdonald symmetric functions with the combinatorics of k-atoms.…”

“…In Section 3, we present four conjecturally equivalent definitions of the k-Schur functions for generic t. Some are known to be equivalent when t = 1. The first definition of k-Schur functions appeared in a paper by Lapointe, Lascoux and Morse [92] and is purely combinatorial in nature; defined as a sum over certain classes of tableaux called atoms. Lapointe and Morse [93] followed this paper by defining symmetric functions which were defined by algebraic operations instead of a sum over combinatorial objects.…”