# Which irreducible representations of the symmetric group are eigenspaces of class sums?

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.