Endomorphism theorem for a regular Specht module
= Endomorphism theorem for a regular Specht module
If $\lambda=(n^{a_n},\ldots,1^{a_1})$ is $p$-regular, then
$$
\operatorname{End}_{\mathbb F S_n}(S^\lambda)=\mathbb F.
$$
For a tableau $t$ and its row reversal $t^*$,
$$
\langle e(t),e(t^*)\rangle=\prod_j(a_j!)^j,
$$
which is nonzero in characteristic $p$. Applying a column antisymmetrizer to an endomorphism at $e(t^*)$ therefore forces its value on the cyclic generator $e(t)$ to be scalar.