Dimension of a representation of $S_n$ Let $Perm_n$ be a permutation representation of $S_n$ (so that $dim=n$), let $V$ be a vector space of dimension $d$ and let $V^{\otimes n}$ be a representation of $S_n$ permuting factors. How can one calculate the dimension of
$$Perm_n \otimes_{S_n} V^{\otimes n},$$
which is a quotient of usual tensor product by an action of $S_n$? I tried to compute the scalar product of the characters of $Perm_n$ and $V^{\otimes n}$, but the combinatorics is overwhelming.
 A: I like $V^{\otimes n} \curvearrowleft S_n$ from the right so we have $(v_1\otimes\cdots\otimes v_n)\sigma= v_{\sigma(1)}\otimes\cdots v_{\sigma(n)}$.
In the $GL(V)$-module $V^{\otimes n}\otimes_{S_n}\mathbb{C}^n$, any pure tensor
$$ (v_1\otimes\cdots\otimes v_n)\otimes(c_1,\cdots,c_n) $$
using the standard basis $\{e_1,\cdots,e_n\}$ of $\mathbb{C}^n$ may be rewritten as
$$\begin{array}{l} \displaystyle 
=(v_1\otimes\cdots\otimes v_n) \otimes \left(\sum_{i=1}^n c_ie_i\right) \\[5pt] \displaystyle =\sum_{i=1}^n c_i \, (v_1\otimes\cdots\otimes v_n)\otimes e_i \\[5pt] \displaystyle =\sum_{i=1}^n c_i \, (v_1\otimes\cdots\otimes v_n)\otimes \sigma_i e_n \\[5pt] \displaystyle =\sum_{i=1}^n c_i \, (v_1\otimes\cdots\otimes v_n)\sigma_i\otimes e_n  \\[5pt] \displaystyle =\sum_{i=1}^n c_i \, (v_{\sigma_{\large i}(1)}\otimes\cdots\otimes v_{\sigma_{\large i}(n)})\otimes e_n \\[5pt] \displaystyle = \left(\sum_{i=1}^n c_i \, v_{\sigma_{\large i}(1)}\otimes\cdots\otimes v_{\sigma_{\large i}(n)} \right)\otimes e_n \end{array} $$
where $\sigma_1,\cdots,\sigma_n\in S_n$ are any permutations for which $\sigma_i e_n=e_i$.
Therefore, the morphism of $GL(V)$-modules $V^{\otimes n}\to V^{\otimes n}\otimes_{S_n}\mathbb{C}^n$ given by $x\mapsto x\otimes e_n$ is an onto map, where $x\in V^{\otimes n}$. The kernel should be comprised of differences $v-\sigma v$ where $\sigma\in S_n$ fixes $e_n$, so heuristically we should expect the module to be the coinvariants,
$$ V^{\otimes n}\otimes_{S_n}\mathbb{C}^n \,\cong\, V^{\otimes n}/S_{n-1} \,\cong\, \mathrm{Sym}^{n-1}(V)\otimes V $$
which has dimension $\displaystyle\binom{d+n-1}{n}d$.
