I have read a lot about idempotents, several important facts were about central idempotents. Now, the Young symmetrizer is a constant away from an idempotent, but I don't think it's central.
Question: What is the centralizer of the Young symmetrizer?
For clarification, let $\lambda\vdash n$ be a partition of $n$ and let $Y$ be the Young diagram of $\lambda$. The symmetric group $S_Y$ has the subgroup $C$ of permutations that leave each column of $Y$ invariant and the subgroup $R$ of permutations that leave the rows invariant. Then, in the group algebra $\mathbb C[S_Y]$, we define elements $$\begin{align*} a_\lambda &:= \sum_{p\in R} p & &\text{and}& b_\lambda &:= \sum_{q\in C} (-1)^q q \end{align*}$$ where $(-1)^q$ is my notation for the signum of a permutation (defined via cycle type). Then, $c_\lambda = a_\lambda \cdot b_\lambda$ is the Young symmetrizer with respect to $\lambda$.