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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\lambda = (\\lambda_1, \\ldots, \\lambda_n)$ be an [[integer partition|0JV30XN73Z]] with at most $n$ nonzero parts, padded with zeros, so $\\lambda_1 \\ge \\cdots \\ge \\lambda_n \\ge 0$.\nThe *Schur polynomial* of $\\lambda$ in the variables $x_1, \\ldots, x_n$ is\n$$s_\\lambda(x_1, \\ldots, x_n) = \\frac{\\det\\big(x_i^{\\lambda_j + n - j}\\big)_{1 \\le i, j \\le n}}{\\det\\big(x_i^{n - j}\\big)_{1 \\le i, j \\le n}}.$$\nHere $\\det$ is the [[determinant|0C3CFHDN6R]], and the denominator is the [[Vandermonde determinant|0P84B554AS]] $\\prod_{i \u003c j} (x_i - x_j)$.\n\n## Remarks\n\nThe numerator is an alternating polynomial, so it is divisible by the Vandermonde determinant, and the quotient is a [[symmetric polynomial|0VHSHB9B6J]] in $\\Z[x_1, \\ldots, x_n]$.\nIt is homogeneous of degree $\\abs{\\lambda} = \\lambda_1 + \\cdots + \\lambda_n$.\n\nExample: for $n = 2$ and $\\lambda = (2, 0)$, the exponents in the numerator are $\\lambda_1 + 1 = 3$ and $\\lambda_2 + 0 = 0$.\nSo $s_{(2)} = \\frac{x_1^3 - x_2^3}{x_1 - x_2} = x_1^2 + x_1 x_2 + x_2^2$.\nFor $\\lambda = (1, 1)$, the same computation gives $s_{(1,1)} = \\frac{x_1^2 x_2 - x_1 x_2^2}{x_1 - x_2} = x_1 x_2$.\n\nNon-example: the sequence $(1, 2)$ is not weakly decreasing, so it is not a partition, and $s_{(1,2)}$ is not a Schur polynomial.\nThe formula would give a numerator with two equal columns, hence $0$.\n\nCombinatorial form: $s_\\lambda(x_1, \\ldots, x_n) = \\sum_T x^T$, a sum over the semistandard [[Young tableaux|Q6TH9PH2R7]] $T$ of shape $\\lambda$ with entries in $\\set{1, \\ldots, n}$.\nHere $x^T = \\prod_i x_i^{t_i}$, where $t_i$ is the number of entries of $T$ equal to $i$.\nSpecial cases are $s_{(k)} = h_k$, the complete homogeneous symmetric polynomial, and $s_{(1, \\ldots, 1)} = e_k$, the [[elementary symmetric polynomial|0X5QVCF06N]] for $k$ ones.\nLetting $n \\to \\infty$ gives the Schur functions in the ring of [[symmetric functions|PAJEHWXW6M]].\n",
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