Expressions for Macdonald polynomials and the generalization of Macdonald's positivity conjecture
We give a determinantal expression for the Macdonald basis which immediately produces efficient recursions for the Macdonald, Jack, and Schur polynomials (Kostka numbers).
We then discuss the long standing open problem of determining statistics
for certain expansion coefficients for the Macdonald polynomials. We present
recent progress that has been made using tableaux combinatorics and in
particular, give a generalization of the Macdonald positivity conjecture.