Specht module

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.