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  "body": "\n## Summary\n\nIn mathematics, the **Young subgroups** of the [[symmetric group|0515A6D2Y1]] $S_n$ are special subgroups that arise in [[combinatorics|0JKW7W7XFR]] and representation theory. When $S_n$ is viewed as the group of [[permutations|05RWFKVTJ7]] of the set $\\{1, 2, \\ldots, n\\}$, and if $\\lambda = (\\lambda_1, \\ldots, \\lambda_\\ell)$ is an [[integer partition|0JV30XN73Z]] of $n$, then the Young subgroup $S_\\lambda$ indexed by $\\lambda$ is defined by\n\n$$S_\\lambda = S_{\\{1, 2, \\ldots, \\lambda_1\\}} \\times S_{\\{\\lambda_1 + 1, \\lambda_1 + 2, \\ldots, \\lambda_1 + \\lambda_2\\}} \\times \\cdots \\times S_{\\{n - \\lambda_\\ell + 1, n - \\lambda_\\ell + 2, \\ldots, n\\}},$$\n\nwhere $S_{\\{a, b, \\ldots\\}}$ denotes the set of permutations of $\\{a, b, \\ldots\\}$ and $\\times$ denotes the [[direct product of groups|00HF4RJ5QN]]. Abstractly, $S_\\lambda$ is isomorphic to the product $S_{\\lambda_1} \\times S_{\\lambda_2} \\times \\cdots \\times S_{\\lambda_\\ell}$. Young subgroups are named for Alfred Young.\n",
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