Let $\lambda \vdash n$. Identify $S^\lambda \otimes sgn$ as a simple representation of $S^n$.
I know that that $S^\lambda$ is the Specht module (over $\mathbb{C}$) with a set of polytabloids as a basis depending on the partition $\lambda$ of $n$. I want to show that $S^\lambda$ is invariant under the action of $S_n$, which shouldn't be too difficult, and that it does not contain a sub-module invariant under the action of $S_n$. This second part is what I'm not sure how to approach. This would give me that $S^\lambda$ would correspond to a simple representation of $S_n$, and when tensored with a 1 dimensional representation, it would give me another simple representation.