Tableau atoms and a new Macdonald positivity conjecture

Abstract

Let $\Lambda$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_\lambda[X/(1-t)]\}_{\lambda_1\leq k}$. We introduce a new family of symmetric polynomials, $\{A_{\lambda}^{(k)}[X;t]\}_{\lambda_1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A_{\lambda}^{(k)}[X;t]$ 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 $A_\lambda^{(k)}[X;t]$ 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 $A_{\lambda}^{(k)}[X;t]$ seem to play the same role for $V_k$ as the Schur functions do for $\Lambda$. 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.