Past exam of the representation Theory of Symmetric Groups course of the University of Oxford / 2011 / 2 / a / Solution

The permutation module $M_F^{\mu }$ is the $F$-vector space with the $\mu$-tabloids as basis, with $S_n$ acting by permuting their entries. For a tableau $t$ with column stabilizer $C_t$, its polytabloid isThe Specht module $S_F^{\mu }$ is the submodule of $M_F^{\mu }$ spanned by these polytabloids, equivalently generated by any one $e_t$ under the group action.

The standard basis theorem states thatis an $F$-basis of $S_F^{\mu }$ in every characteristic. Its dimension is the number of standard tableaux of shape $\mu$; in the hook shape used below the basis is exactly the displayed $e_{1ij}$ with $2\le i<j\le n$.