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A. Garsia and G. Tesler,
Plethystic Formulas for Macdonald q,t-Kostka Coefficients,
Advances in Mathematics, 123 no. 2 (1996), 143-222.

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Abstract

This work is concerned with the Macdonald q,t-analogue of the Kostka matrix. This matrix relates the two parameter Macdonald basis {P\mu(x;q,t)} to the modified Schur basis {S\lambda[X(1-t)]}. The entries in this matrix, which have come to be denoted by K\lambda,\mu(q,t), have been conjectured by Macdonald to be polynomials in q,t with positive integral coefficients. Our main main result here is an algorithm for the construction of explicit formulas for the K\lambda,\mu(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 do also yield positivity of the coefficients for a wide variety of entries in the Macdonald q,t-Kostka matrix. We also obtain in this manner new explicit expressions for the Macdonald polynomials.