Symmetric invariants of a Schur Module

Question

Let $V\cong\mathbb C^n$ be a complex vector space of dimension $n$. Let $\lambda\in\mathbb Z^r$ be a generalized integer partition $\lambda_1\ge\cdots\ge\lambda_r$ with $r\le n$. Denote by $\mathbb S_\lambda(V)$ the Schur module with respect to $\lambda$, i.e. the irreducible $\operatorname{GL}(V)$ representation of weight $\lambda$. Consider the symmetric group $\mathfrak S_n$ acting on $V$ by permutation of the basis vectors. This action lifts to an action on $\mathbb S_\lambda(V)$.

I know that the decomposition of the zero weight space $\mathbb S_\lambda(V)_0$ (with respect to a max. torus of $\operatorname{GL}(V)$) into $\mathfrak S_n$-modules has certain plethysmic coefficients, but how about the decomposition of $\mathbb S_\lambda(V)$ into $\mathfrak S_n$-modules? Is it known? Most importantly, I would like to know when $\mathbb S_\lambda(V)$ contains a (nonzero) $\mathfrak S_n$ invariant.

Thanks in advance already!

Answer 1

The symmetric group action you have described is exactly the action of the weyl group on the weight spaces. Let $v$ be a highest weight vector. Then $$ \sum_{\sigma \in S_n} \sigma v $$ is non zero and invariant under the symmetric group action. The reason it is non zero is because the component in the highest weight subspace is non zero.