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  "body": "\n## Definition\n\nLet $\\Lambda$ be the ring of [[symmetric functions|PAJEHWXW6M]] in variables $x = (x_1, x_2, \\dots)$ with coefficients in the field $\\Q(q, t)$.\nFor a [[partition|0JV30XN73Z]] $\\lambda$, let $m_\\lambda$ be the monomial symmetric function and let $p_\\lambda = p_{\\lambda_1} p_{\\lambda_2} \\cdots$ be the product of [[power sums|0ZNXZB4HNB]].\nLet $z_\\lambda = \\prod_{i \\ge 1} i^{m_i} m_i!$, where $m_i$ is the number of parts of $\\lambda$ equal to $i$.\nDefine a bilinear form on $\\Lambda$ by\n$$\\ip{p_\\lambda}{p_\\mu}_{q,t} = \\delta_{\\lambda\\mu} \\, z_\\lambda \\prod_{i} \\frac{1 - q^{\\lambda_i}}{1 - t^{\\lambda_i}}.$$\nFor partitions $\\lambda, \\mu$ of the same number, write $\\mu \\le \\lambda$ (dominance order) if $\\mu_1 + \\dots + \\mu_k \\le \\lambda_1 + \\dots + \\lambda_k$ for all $k$.\nThe *Macdonald polynomials* are the unique family $(P_\\lambda(x; q, t))_\\lambda$ in $\\Lambda$ such that:\n\n1. $P_\\lambda = m_\\lambda + \\sum_{\\mu \u003c \\lambda} c_{\\lambda\\mu} m_\\mu$ for some $c_{\\lambda\\mu} \\in \\Q(q, t)$.\n2. $\\ip{P_\\lambda}{P_\\mu}_{q,t} = 0$ for $\\lambda \\ne \\mu$.\n\n## Remarks\n\nThe existence and uniqueness of this family is a theorem of Macdonald; it is not automatic, because dominance is only a partial order.\n\nExample: for $\\lambda = (2)$, $P_{(2)} = m_{(2)} + \\frac{(1+q)(1-t)}{1-qt} m_{(1,1)}$.\nCheck: $m_{(2)} = p_2$ and $m_{(1,1)} = \\tfrac12 (p_1^2 - p_2)$, and $z_{(2)} = z_{(1,1)} = 2$.\nThen $\\ip{p_2}{m_{(1,1)}}_{q,t} = -\\frac{1-q^2}{1-t^2}$ and $\\ip{m_{(1,1)}}{m_{(1,1)}}_{q,t} = \\frac{(1-q)(1-qt)}{(1-t)(1-t^2)}$, so $\\ip{P_{(2)}}{P_{(1,1)}}_{q,t} = 0$ with $P_{(1,1)} = m_{(1,1)}$.\n\nNon-example: the Schur function $s_{(2)} = m_{(2)} + m_{(1,1)}$ satisfies clause 1 but not clause 2 when $q \\ne t$.\nIndeed $\\ip{s_{(2)}}{m_{(1,1)}}_{q,t} = \\frac{(1-q)(t-q)}{(1-t)(1-t^2)} \\ne 0$.\n\nAt $q = t$ the form is the Hall inner product and $P_\\lambda = s_\\lambda$.\nAt $q = 0$, $P_\\lambda$ is the Hall–Littlewood polynomial $P_\\lambda(x; t)$.\n",
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