Plethystic Formulas for Macdonaldq, t-Kostka Coefficients

AbstractThis work is concerned with the Macdonaldq, t-analogue of the Kostka matrix. This matrix relates the two parameter Macdonald basis {Pμ(x; q, t)} to the modified Schur basis {Sλ[X(1−t)]}. The entries in this matrix, which have come to be denoted byKλ, μ(q, t), have been conjectured by Macdonald to be polynomials inq, twith positive integral coefficients. Our main result here is an algorithm for the construction of explicit formulas for theKλ, μ(q, t). It is shown that this algorithm yields expressions which are polynomials with integer coefficients. Recent work of J. Remmel shows that the resulting formulas also yield positivity of the coefficients for a wide variety of entries in the Macdonaldq, t-Kostka matrix. We also obtain in this manner new explicit expressions for the Macdonald polynomials