Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 1 / a / i / Solution

The symmetric group acts transitively on the tableaux of a fixed shape. Thus every $s\in {\Delta }^{\lambda }$ is $gt$ for some $g\in S_n$, and the definition of a polytabloid givesSince the Specht module is spanned by all $e(s)$, it is the cyclic $\mathbb{F}{S}_n$-module generated by $e(t)$.