The tensor product of two complex representations

Let $\rho :G\to \mathrm{GL}(V)$ and $\sigma :G\to \mathrm{GL}(W)$ be finite-dimensional complex representations of a group $G$ (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree). Their tensor product representation $\rho \otimes \sigma$ is the representation of $G$ on the complex tensor product $V{\otimes }_{\mathbb{C}}W$ given on elementary tensors by

$g\cdot (v\otimes w):=(\rho (g)v)\otimes (\sigma (g)w)(g\in G,v\in V,w\in W)\mathrm{.}$

The action is well defined. For fixed $g\in G$, the assignment $(v,w)\mapsto \rho (g)v\otimes \sigma (g)w$ is additive in each variable and $\mathbb{C}$-balanced, because $\rho (g)$ and $\sigma (g)$ are $\mathbb{C}$-linear: $\rho (g)(v\lambda )\otimes \sigma (g)w=(\rho (g)v)\lambda \otimes \sigma (g)w$. By Universal property of the tensor product for balanced maps into abelian groups there is a unique $\mathbb{C}$-linear map $\rho (g)\otimes \sigma (g):V{\otimes }_{\mathbb{C}}W\to V{\otimes }_{\mathbb{C}}W$ with the displayed value on every elementary tensor. Because the two sides of $g\cdot (h\cdot (v\otimes w))=(gh)\cdot (v\otimes w)$ agree on every elementary tensor and the elementary tensors span $V{\otimes }_{\mathbb{C}}W$, the assignment $g\mapsto \rho (g)\otimes \sigma (g)$ is a group homomorphism into $\mathrm{GL}(V{\otimes }_{\mathbb{C}}W)$, with $\rho (1)\otimes \sigma (1)=\mathrm{id}$. The underlying space is finite-dimensional: bases of $V$ and $W$ give the basis $(v_i\otimes w_j{)}_{i,j}$, so $\dim (V{\otimes }_{\mathbb{C}}W)=(\dim V)(\dim W)$.