---
tag: D8KET8700Y
kind: definition
name: Specht module
msc: [20C30]
defines: NTFM7X5555
uses: [0515A6D2Y1, 0JV30XN73Z, 096AHMB3P4, Q6TH9PH2R7, 02DKV69ZH0, MNP2HBWB5Q, 0X1JXT47GA]
sources:
  - url: https://en.wikipedia.org/wiki/Specht_module
---

## Definition

Let $n \ge 1$, let $F$ be a field, and let $\lambda$ be a [[partition|0JV30XN73Z]] of $n$ with [[Young diagram|096AHMB3P4]] $D_\lambda$.
A [[Young tableau|Q6TH9PH2R7]] $t$ of shape $\lambda$ is a bijective filling of the boxes of $D_\lambda$ with $1, \ldots, n$.
The [[symmetric group|0515A6D2Y1]] $S_n$ acts on tableaux by acting on the entries.
Two tableaux are row-equivalent if corresponding rows have the same sets of entries.
The class $\set{t}$ of $t$ under this relation is a tabloid.
Let $M^\lambda$ be the vector space over $F$ with basis the tabloids of shape $\lambda$, with the induced action of $S_n$.
Let $C_t \le S_n$ be the column [[subgroup|0X1JXT47GA]] of $t$, the permutations that keep each column of $t$ as a set.
The polytabloid of $t$ is
$$e_t = \sum_{\sigma \in C_t} \sgn(\sigma) \, \sigma \set{t} \in M^\lambda,$$
where $\sgn$ is the [[sign|02DKV69ZH0]].
The *Specht module* $S^\lambda$ is the subspace of $M^\lambda$ spanned by the $e_t$ for all tableaux $t$ of shape $\lambda$.

## Remarks

Since $\pi e_t = e_{\pi t}$ for $\pi \in S_n$, the space $S^\lambda$ is a representation of $S_n$, that is, an $F[S_n]$-module.
Example: for $\lambda = (n)$, one row, there is only one tabloid and $C_t$ is trivial, so $S^{(n)}$ is the trivial representation of degree $1$.
For $\lambda = (1^n)$, 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.
Non-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 $n - 1$.
The polytabloids $e_t$ for standard tableaux $t$, increasing along rows and down columns, form a basis of $S^\lambda$.
If $F$ 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.