Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$.
Let $\lambda$ be a Young diagram of size $n$ and denote by $V_\lambda$ the corresponding Specht module (irreducible representation of $S_n$ corresponding to $\lambda$).
Since $c_a \in \mathbb{C}[S_n]$ is central it must act on $V_{\lambda}$ by some scalar $c_a(\lambda)$.
Question: are there any “simple” formulas for these scalars $c_a(\lambda)$?