Let $T$ be a Young tableaux (rows of decreasing length, filled by integers $1,2,\ldots,n$), of shape $\lambda=(\lambda_1,\cdots,\lambda_k)$
Let $P$ be the subgroup of $S_n$ consisting of permutations preserving rows of $T$.
In the group algebra $\mathbb{C}S_n$, consider the element $$ a=\sum_{\sigma\in P}\sigma. $$
$S_n$ acts on $V^{\otimes n}$ by permuting factors; the image of element $(a\in \mathbb{C}S_n)\rightarrow \mathrm{End}(V^{\otimes n})$ is just the subspace $$ Im(a)=\mathrm{Sym}^{\lambda_1}V \otimes \mathrm{Sym}^{\lambda_2}V \otimes\cdots \otimes \mathrm{Sym}^{\lambda_k}V \subset V^{\otimes n}$$ where the inclusion on the right is obtained by grouping the factors of $V^{\otimes n}$ according to the rows of tableaux $T$.
Question: It may be trivial, but I do not understand the above statement (from Fulton-Harris, Rep. Theory). In the endomorphism ring, I could not get how $Im(a)$ is subspace of the form mentioned above? Can one explain it?
