Let $S_n$ be the symmetric group on $n$ symbols and let $\lambda = (\lambda_1,\dots,\lambda_r)$ be a partition of $n$. Considering the Young diagram associated to this partition we can define the subgroups $P$ and $Q$ of $S_n$ whose elements respectively preserve each row and each column of the diagram.
In that case if $\mathbb{C}S_n$ is the group algebra we define
$$a_\lambda = \sum_{g\in P}e_g,\quad b_\lambda = \sum_{g\in Q}\operatorname{sgn}(g) e_g,$$
we also define $c_\lambda = a_\lambda b_\lambda$ the associated Young symmetrizer.
Now let $V$ be a vector space and suppose $S_n$ acts on $V^{\otimes n}$ by permutating the factors. We can extend this action to allow $c_\lambda$ act on $V^{\otimes n}$ and thus we get one representation of $\mathbb{C}S_n$ on $V^{\otimes n}$.
Thus if we take the partition $\lambda = (n)$ we have $c_\lambda = \sum_{g\in S_n} e_g$ and hence
$$c_\lambda(V^{\otimes n})=\operatorname{Sym}^n V.$$
I have read that if $[\lambda_1,\dots,\lambda_r]$ denotes the irreducible of the symmetric group representation associated to the partition $(\lambda_1,\dots,\lambda_r)$, we then have the decomposition into irreducible parts
$$[n]\otimes [1]=[n,1]\oplus [n+1],$$
which in index notation decomposes one $F^{a_1\dots a_n a_{n+1}}$ symmetric on the first $n$ indices into the parts $F_1^{a_1\dots a_n a_{n+1}}$ and $F_2^{a_1\dots a_{n} a_{n+1}}$ where $F_1$ is totally symmetric and $F_2$ is antisymmetric on the last two indices.
Now I want to understand this and understand how to find this kind of decomposition in general.
I believe the point is: $[n]$ corresponds to the representation associated with the partition $(n)$ and $[1]$ that with partition $(1)$. These give symmetrizers $c_{(n)},c_{(1)}$ acting respectively on $V^{\otimes n}$ and $V$. Thus we get one symmetrizer $c_{(n)}\otimes \mathbf{1}+\mathbf{1}\otimes c_{(1)}$ acting on $V^{\otimes {n+1}}$ such that
$$c_{(n)}\otimes \mathbf{1}+\mathbf{1}\otimes c_{(1)}(V^{\otimes n}\otimes V)=\operatorname{Sym}^n V\otimes V$$
and this is decomposed.
Now my doubts are:
Which representations are being dealt with here? Representations of $S_n$ or $\mathbb{C}S_n$? I believe it is of the later.
How one decomposes $[n]\otimes [1]=[n,1]\otimes [n+1]$ in practice? I believe this has something to do with the so-called Weyl construction, but I'm totally unsure.
