I have some problems understanding a decomposition used by Fulton and Harris in their book on representation theory (and also Procesi in his Lie group's book).
Given a partition of $d$, i.e. a vector $\lambda=(\lambda_1,\ldots,\lambda_k)$ where the $\lambda_i$ are non negative integers and $\lambda_1+\ldots+\lambda_k=d$. Define the usual Young diagram and inserting integer $1,\ldots,d$ in this diagram you get a tableau. Define
\begin{align*}
P_\lambda=\{g\in S_d\,|\,g\;\,\text{preserve the row} \},\quad Q_\lambda=\{g\in S_d\,|\,g\;\,\text{preserve the column} \}.
\end{align*}
Then we have
\begin{align*}P_\lambda=S_{\lambda_1}\times\ldots\times S_{\lambda_k},\;\;\text{and}\;\;Q_\lambda=S_{n_1}\times\ldots\times S_{n_k}\end{align*}
where in the latter decomposition $n_i$ is the length of the $i$-th column.
I don't understand first why this is a direct product, each $S_{\lambda_i}$ and $S_{\lambda_i+1}$ doesn't intersect trivially (idem for the $n_i$). But also i don't think this is the right decomposition.
For the second question I think that maybe they label a tableau in this way: $1,\ldots,\lambda_1$ in the first row, $1,\ldots,\lambda_2$ in the second row, etc (specular for the column), but the fist problem is still unsolved.
Any help? Thanks.