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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $F$ be a field, and let $\\lambda$ be a [[partition|0JV30XN73Z]] of $n$ with [[Young diagram|096AHMB3P4]] $D_\\lambda$.\nA [[Young tableau|Q6TH9PH2R7]] $t$ of shape $\\lambda$ is a bijective filling of the boxes of $D_\\lambda$ with $1, \\ldots, n$.\nThe [[symmetric group|0515A6D2Y1]] $S_n$ acts on tableaux by acting on the entries.\nTwo tableaux are row-equivalent if corresponding rows have the same sets of entries.\nThe class $\\set{t}$ of $t$ under this relation is a tabloid.\nLet $M^\\lambda$ be the vector space over $F$ with basis the tabloids of shape $\\lambda$, with the induced action of $S_n$.\nLet $C_t \\le S_n$ be the column [[subgroup|0X1JXT47GA]] of $t$, the permutations that keep each column of $t$ as a set.\nThe polytabloid of $t$ is\n$$e_t = \\sum_{\\sigma \\in C_t} \\sgn(\\sigma) \\, \\sigma \\set{t} \\in M^\\lambda,$$\nwhere $\\sgn$ is the [[sign|02DKV69ZH0]].\nThe *Specht module* $S^\\lambda$ is the subspace of $M^\\lambda$ spanned by the $e_t$ for all tableaux $t$ of shape $\\lambda$.\n\n## Remarks\n\nSince $\\pi e_t = e_{\\pi t}$ for $\\pi \\in S_n$, the space $S^\\lambda$ is a representation of $S_n$, that is, an $F[S_n]$-module.\nExample: for $\\lambda = (n)$, one row, there is only one tabloid and $C_t$ is trivial, so $S^{(n)}$ is the trivial representation of degree $1$.\nFor $\\lambda = (1^n)$, one column, each row has a single box, so there are $n!$ tabloids, $C_t = S_n$ and $\\pi e_t = \\sgn(\\pi) e_t$, so $S^{(1^n)}$ is the sign representation.\nNon-example: the space $M^\\lambda$ itself is not in general a Specht module; for $\\lambda = (n-1, 1)$ it is the permutation representation on $n$ points, of dimension $n$, while $S^{(n-1,1)}$ has dimension $n - 1$.\nThe polytabloids $e_t$ for standard tableaux $t$, increasing along rows and down columns, form a basis of $S^\\lambda$.\nIf $F$ has characteristic $0$, the modules $S^\\lambda$ are irreducible, pairwise non-isomorphic, and every irreducible representation of $S_n$ over $F$ is isomorphic to one of them.\n",
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