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 $$\varphi_1 \otimes \varphi_2$$.