Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 1 / c / i / Solution

The row stabilizer of the transposed tableau is $R(t^{\prime })=C(t)$. For $r\in R(t^{\prime })$, the polytabloid satisfiesThe two signs cancel in the tensor product, soThus the proposed value depends only on the tabloid $\{g{t}^{\prime }\}$ and $\theta$ is well-defined. Its definition immediately givesso it is an $F{S}_n$-homomorphism. Since any $e(w)$ is $ge(t)$ for some $g$, its images contain every generator $e(w)\otimes e(u)$ of $S^{\lambda }\otimes S^{(1^n)}$. Hence $\theta$ is surjective.