TABLEAU ATOMS AND A NEW MACDONALD POSITIVITY

Let3 be the space of symmetric functions, and let V k be the subspace spanned by the modified Schur functions {Sλ[X/(1−t)]}λ1≤k. We introduce a new family of symmetric polynomials,{A λ [X; t]}λ1≤k, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials A (k) λ [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) λ [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) λ [X; t] seem to play the same role for V k as the Schur functions do for 3. 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.