= 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> is
$$
e_t=\sum_{g\in C_t}\operatorname{sgn}(g)\{gt\}.
$$
The <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 that
$$
\boxed{\{e_t:t\text{ is a standard }\mu\text{-tableau}\}}
$$
is an $F$-basis of $S_F^\mu$ in every <characteristic>. Its <dimension> is the number of <standard tableaux> of shape $\mu$; in the <hook of a Young diagram>[hook] shape used below the <basis> is exactly the displayed $e_{1ij}$ with $2\leq i<j\leq n$.
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