Abstract

Let ? be the space of symmetric functions, and let Vk be the subspace spanned by the modified Schur functions {S? [X/(1-t)]}?1?k. We introduce a new family of symmetric polynomials, {A?(k) [X; t]}?1?k, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials A?(k)[X; t] form a basis for Vk 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 also substantially refine it. Our construction of the A?(k)[X; t] relies on the use of tableau 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) [X; t] seem to play the same role for Vk as the Schur functions do for ?. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri-type and Littlewood-Richardson-type coefficients.