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  "body": "\n## Summary\n\nMacdonald polynomials $P_\\lambda(x; q, t)$ are [[symmetric functions|PAJEHWXW6M]] with coefficients in the field $\\Q(q, t)$ of rational functions in two parameters, indexed by [[partitions|0JV30XN73Z]].\nThey are characterized by a triangularity condition and orthogonality for a two-parameter inner product.\nSpecial values and limits of $q$ and $t$ give the [[Schur polynomials|43VFMJCDYJ]], the Hall–Littlewood polynomials and the Jack polynomials.\nMacdonald introduced them in the 1980s, and they connect combinatorics with representation theory and special functions.\n\n## Explanations\n\n### L1 human\n\nMany classical families of symmetric polynomials are special cases of one family with two extra parameters $q$ and $t$.\nSetting $q = t$ gives back the Schur polynomials, and setting $q = 0$ gives the Hall–Littlewood polynomials.\nFor example, for the partition $(2)$ the Macdonald polynomial is $m_{(2)} + \\frac{(1+q)(1-t)}{1-qt} m_{(1,1)}$, where $m_{(2)} = \\sum_i x_i^2$ and $m_{(1,1)} = \\sum_{i\u003cj} x_i x_j$.\nAt $q = t$ the fraction equals $1$, and the result is the Schur polynomial $s_{(2)}$.\n",
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