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In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\chi_V(C)/\dim(V)$ with $\chi_V$ the character of $V$.

My question is which irreducible representations $V$ of the symmetric group are eigenspaces of a class sum $1_C$. That is, when does there exists a conjugacy class $C$ such that $\lambda(C,W)\neq \lambda(C,V)$ for all $W\neq V$?

If this holds, then the isotypic component corresponding to $V$ in any representation is an eigenspace of $1_C$. A simple example is when $V$ is the $2$-dimensional irreducible representation of $\mathrm S_3$ and $C$ the class of $3$-cycles. Then $V$ is the eigenspace of $1_C$ with eigenvalue $-1$, or equivalently the kernel of $\operatorname{id}+(123)+(132)$. If we let $\mathrm S_3$ act on trilinear forms, this means that the corresponding isotypic subspace consists of forms satisfying $$f(u,v,w)+f(v,w,u)+f(w,u,v)=0. $$ Similarly, if we can do this for an irreducible representation $V$, then we can describe the corresponding Schur functor on multilinear forms as the solution space to a rather simple equation.

I checked using Maple that all irreducible representations of $\mathrm S_n$ for $n\leq 8$ are eigenspaces in the sense explained. When $n\leq 5$ and $n=7$ there is even a conjugacy class $C$ that works for all $V$, but for $n=6$ and $n=8$ that is not the case.

Edit: Perhaps it is worth mentioning that a more standard equation describing an isotypic component is $$\sum_C\overline{\chi_V(C)}|C|\pi(1_C)v=\frac{|G|}{\dim V}\,v. $$ The question is about when the sum on the left can be replaced by a single term.

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