Outer tensor product of linear representations

Definition

Suppose $G_{1},G_{2}$ are groups and $K$ is a field. Suppose $\rho _{1}:G_{1}\to GL(V_{1})$ and $\rho _{2}:G_{2}\to GL(V_{2})$ are linear representations of $G_{1},G_{2}$ respectively over $K$. The outer tensor product, denoted $\rho _{1}\boxtimes \rho _{2}$, is a linear representation of $G_{1}\times G_{2}$ on the tensor product of vector spaces $V_{1}\otimes _{K}V_{2}$ defined in the following equivalent ways.

Direct definition in terms of tensor product of vector spaces

The outer tensor product representation is defined as follows:

$(\rho _{1}\otimes \rho _{2})(g_{1},g_{2})=\rho _{1}(g_{1})\otimes \rho _{2}(g_{2})$

The $\otimes$ on the right is the natural homomorphism:

$GL(V_{1})\times GL(V_{2})\to GL(V_{1}\otimes V_{2})$

Definition in terms of tensor product of linear representations

Let $\pi _{1}:G_{1}\times G_{2}\to G_{1}$ and $\pi _{2}:G_{1}\times G_{2}\to G_{2}$ be the projection maps onto the direct factors. Let $\varphi _{1}=\rho _{1}\circ \pi _{1}$ and $\varphi _{2}=\rho _{2}\circ \pi _{2}$. Then, both $\varphi _{1}$ and $\varphi _{2}$ are linear representations of $G_{1}\times G_{2}$. The outer tensor product of $\rho _{1}$ and $\rho _{2}$ is defined as the tensor product of linear representations $\varphi _{1}\otimes \varphi _{2}$.