$V=\mathbb C^n$ 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 $S_n\times S_m$.