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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer, let $S_n$ be the [[symmetric group|0515A6D2Y1]] on $\\set{1, \\ldots, n}$, and let $F$ be a field.\nA *representation of the symmetric group* $S_n$ over $F$ is a [[group representation|WYGRXCG2RN]] $\\rho \\colon S_n \\to GL(V)$ with $V$ a vector space over $F$.\nThe *representation theory of the symmetric group* is the study of these representations, of their decomposition into [[irreducible representations|FX3NEFYFFN]], and of their characters.\n\n## Remarks\n\nExample: the sign map $\\sgn \\colon S_n \\to \\set{\\pm 1} \\subseteq GL_1(\\C)$ is a representation of degree $1$, because $\\sgn(\\sigma\\tau) = \\sgn(\\sigma)\\sgn(\\tau)$.\nNon-example: let $\\rho \\colon S_3 \\to GL_1(\\C)$ send $1$ to $1$, each transposition to $-1$ and each $3$-cycle to $\\omega = e^{2\\pi i/3}$.\nIt is not a representation: the product of the transpositions $(1\\,2)$ and $(2\\,3)$ is a $3$-cycle, while $(-1)(-1) = 1 \\ne \\omega$.\nOver $\\C$, the number of irreducible representations of $S_n$ is the number of [[conjugacy classes|0DEBXAJ5SA]] of $S_n$.\nConjugacy classes correspond to cycle types, so this is the number $p(n)$ of [[partitions|0JV30XN73Z]] of $n$.\nThe Specht modules $S^\\lambda$, for $\\lambda$ a partition of $n$, are a complete list of irreducible complex representations, without repetition.\nOver a field of characteristic $p$ with $p \\le n$, the theory is the modular representation theory of $S_n$, and is much less understood.\n",
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