For a partition $\lambda$ let $S^{\lambda}$ be the corresponding Schur functor. Is it true that for every $\lambda$ there exists an irreducible representation $V$ of a finite group $G$ such that $S^{\lambda}(V)$ is still irreducible?

This is not obvious to me even for the symmetric and exterior powers (although maybe I'm not thinking hard enough), so any partial results would be appreciated.

For a partition $\lambda$ let $S^{\lambda}$ be the corresponding Schur functor. Is it true that for every $\lambda$ there exists an irreducible representation $V$ of a finite nonabelian group $G$ such that $S^{\lambda}(V)$ is still irreducible?

This is not obvious to me even for the symmetric and exterior powers (although maybe I'm not thinking hard enough), so any partial results would be appreciated.

Let $V = \mathbb{C}^n$ be the standard representation of $\text{GL}_n(\mathbb{C})$, and let $f(n)$ be the largest value of $k$ such that there exists a finite subgroup $G\subset \text{GL}_n(\mathbb{C})$ with the property that every irreducible subrepresentation $W$ of $V^{\otimes k}$ remains irreducible after restriction to $G$. What's known about the function $f$? In particular, is it unbounded?

For example, I think (but am not sure) that $f(2) = 4$, where the finite subgroup is the binary icosahedral group.

For a partition $\lambda$ let $S^{\lambda}$ be the corresponding Schur functor. Is it true that for every $\lambda$ there exists an irreducible representation $V$ of a finite group $G$ such that $S^{\lambda}(V)$ is still irreducible?

This is not obvious to me even for the symmetric and exterior powers (although maybe I'm not thinking hard enough), so any partial results would be appreciated.

Let $V = \mathbb{C}^n$ be the standard representation of $\text{GL}_n(\mathbb{C})$, and let $f(n)$ be the largest value of $k$ such that every irreducible subrepresentation $W$ of $V^{\otimes k}$ remains irreducible after restriction to some finite subgroup of $\text{GL}_n(\mathbb{C})$. What's known about the function $f$? In particular, is it unbounded?

For example, I think (but am not sure) that $f(2) = 4$, where the finite subgroup is the binary icosahedral group.

Let $V = \mathbb{C}^n$ be the standard representation of $\text{GL}_n(\mathbb{C})$, and let $f(n)$ be the largest value of $k$ such that there exists a finite subgroup $G\subset \text{GL}_n(\mathbb{C})$ with the property that every irreducible subrepresentation $W$ of $V^{\otimes k}$ remains irreducible after restriction to $G$. What's known about the function $f$? In particular, is it unbounded?

For example, I think (but am not sure) that $f(2) = 4$, where the finite subgroup is the binary icosahedral group.