I am trying to wrap my head around some things involving representations of symmetric groups in arbitrary characteristic, and some clarification on basic points would be welcome.
Let $V$ be a (finite-dimensional) $K[\mathfrak{S}_n]$-module, where $K$ is any field and $n>1$. To simplify notations I will not make a distinction between an element of $K[\mathfrak{S}_n]$ and the map it induces on $V$ (it should not introduce any confusion). Consider the "antisymmetrizing" element $c_n = \sum_{\sigma\in \mathfrak{S}_n}\epsilon(\sigma)\sigma\in K[\mathfrak{S}_n]$.
Then on one side we get $$\sum_\tau \ker(1-\tau)\stackrel{(1)}{\subseteq} \ker(c_n) $$ where the sum is taken over the transpositions $\tau$, and on the other side $$ c_n V \stackrel{(2)}{\subseteq} \bigcap_\tau (1-\tau)V. $$
I am struggling to see exactly when those inclusions are equalities. Of course if the characteristic of $K$ does not divide $n!$ (so it is either $0$ or some $p>n$) then there is no problem, we are in the semi-simple case, $c_\lambda$ is (up to a non-zero scalar) the canonical projection on the isotypical component of $V$ corresponding to the sign representation, and both equalities hold. Also, when $V=U^{\otimes n}$ with the natural action of $\mathfrak{S}_n$, it seems to me that both inclusions are equalities regardless of the characteristic (and $c_n V \simeq \Lambda^n(U)$).
What is the general situation? If the equalities do not always hold, is there a natural condition which ensures that they do?