What is the centralizer of the Young symmetrizer? 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 $\lambda$.
 A: Since the Young symmetrizer is only defined upto conjugation by an element in the group basis, it is far from central! Lets be precise.
Let $ \lambda $ be a partition of $ n $. A numbering $T$ of $ \lambda$ is just a way of putting the numbers $ 1 $ to $n$ in the boxes of $ \lambda $ without repetition. The group $S_n$ act on the numberings of $ \lambda $ in the obvious way. We define $R(T) $ to be the permutations which stabilize the rows of $T$ and $C(T)$ to be the permutations which stabilize the columns of $T$. We then define the row symmetrizer and column anti-symmetrizer by
$$ a_{T} = \sum_{\sigma \in R(T)} \sigma $$
$$ b_{T} = \sum_{\sigma \in C(T)} (-1)^{\sigma} \sigma $$
The young symmetrizer is then defined by $ c_T = a_T b_T $. We also have the formula $$ \sigma c_T \sigma^{-1} = c_{\sigma \cdot T} $$
Since $S_n$ acts transitively on the numberings of $ \lambda$, modifying $ T $ is the same as hitting $c_T$ with an inner automorphism. We don't really care about how the symmetric group is labeled, so we abuse notation and define $ c_{\lambda}$, the young symmetrizer, which is only defined upto inner automorphism. 
One could still ask about the centralizer of a specific $ c_T$, but there is a sense in which this isn't so important: indeed, the irreps of $S_n$ are parameterized by partitions, not numberings of partitions.
