Macdonald polynomial

Summary

Macdonald polynomials $P_\lambda(x; q, t)$ are symmetric functions with coefficients in the field $\Q(q, t)$ of rational functions in two parameters, indexed by partitions. They are characterized by a triangularity condition and orthogonality for a two-parameter inner product. Special values and limits of $q$ and $t$ give the Schur polynomials, the Hall–Littlewood polynomials and the Jack polynomials. Macdonald introduced them in the 1980s, and they connect combinatorics with representation theory and special functions.