  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.
