$V=\mathbb C^n$ is a $\mathbb CS\_n$-module, where $S\_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^{\otimes m}$ is therefore also a $\mathbb CS\_n$-module via the action $\sigma(v\_1\otimes\cdots\otimes v\_m) = \sigma v\_1\otimes\cdots \otimes \sigma v\_m$ on elementary tensors. 


But $V^{\otimes m}$ is also a $\mathbb CS\_m$-module where $S\_m$ acts by permuting the tensor factors. These two actions commute and hence $V^{\otimes m}$ is a representation of $S\_n\times S\_m$ in a natural way. I would like a pointer to the literature on the decomposition of $V^{\otimes m}$ into irreducible representations of $S\_n\times S\_m$.



Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$
By Schur–Weyl duality there is an isomorphism of $\mathrm{GL}(V) \times S\_m$-representations


$$V^{\otimes m} \cong \bigoplus\_\lambda \Delta^\lambda(V) \boxtimes S^\lambda$$


where the sum is over all partitions $\lambda$ of $m$ with at most $n$ parts, $\Delta^\lambda$ is the Schur functor for $\lambda$, $S^\lambda$ is the irreducible $\mathbb{C}S\_m$-module canonically labelled by $\lambda$ and $\boxtimes$ denotes the outer tensor product of two representations. Restricting to $S\_n \times S\_m$ we get


$$V^{\otimes m} \cong \bigoplus\_\lambda \bigl( \Delta^\lambda(V) \bigl\downarrow^{\mathrm{GL}(V)}\_{S\_n} \bigr) \boxtimes S^\lambda.$$


Hence 


$$[V^{\otimes m} : S^\nu \boxtimes S^\lambda] = 
[\Delta^\lambda(V) : S^\nu]$$


for all partitions $\nu$ of $n$ and $\lambda$ of $m$ with at most $n$ parts.


Determining the multiplicities on the right-hand side is, as far as I know, an open problem, equivalent to computing [inner plethysms](https://mathoverflow.net/questions/96705/computer-package-for-representation-theory-of-the-symmetric-group) of symmetric functions. Some special cases are known. For instance, if $\lambda = (1^m)$ and $m \le n$ then $\Delta^{(1^m)}(V) = \bigwedge^mV = S^{(n-m,1^m)} \oplus S^{(n-m+1,1^{m-1})}$. This result has been [reproved](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=RVCN&pg8=ET&review_format=html&s4=hamernik&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=404408), [many](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=RVCN&pg8=ET&review_format=html&s4=muller&s5=vertices&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=4&mx-pid=2341720), [times (see Proposition 5.1)](http://arxiv.org/pdf/1411.3502.pdf)  in the symmetric group literature.


