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  "body": "\n## Summary\n\nA symmetric function is a formal power series of bounded degree in infinitely many variables $x_1, x_2, \\ldots$, see the [[definition|NS852SSQ00]].\nIt does not change under any permutation of the variables.\nThe symmetric functions form the ring $\\Lambda$, which collects the [[symmetric polynomials|0VHSHB9B6J]] in all numbers of variables at once.\nThe ring has several bases indexed by [[integer partitions|0JV30XN73Z]]: monomial, [[elementary|0X5QVCF06N]], complete homogeneous and [[Schur functions|43VFMJCDYJ]].\nOver $\\Q$, the [[power sum|0ZNXZB4HNB]] symmetric functions give one more basis.\nIt is a central object of [[algebraic combinatorics|VPAQZ0XW74]] and of the representation theory of symmetric and general linear groups.\n\n## Explanations\n\n### L1 human\n\nA symmetric polynomial, such as $x_1^2 + x_2^2 + x_3^2$, stays the same when the variables are shuffled.\nIdentities between symmetric polynomials usually do not depend on the number of variables.\nSo one works with infinitely many variables at once, as in $x_1^2 + x_2^2 + x_3^2 + \\cdots$.\nThese infinite sums of bounded degree are the symmetric functions.\nSetting all but finitely many variables to $0$ gives back the symmetric polynomials.\n",
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