Outer tensor product of linear representations

Definition

Suppose $G_{1}, G_{2}$ are groups and $K$ is a field. Suppose $ρ_{1} : G_{1} \to G L (V_{1})$ and $ρ_{2} : G_{2} \to G L (V_{2})$ are linear representations of $G_{1}, G_{2}$ respectively over $K$ . The outer tensor product, denoted $ρ_{1} ⊠ ρ_{2}$ , 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:

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

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

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

Definition in terms of tensor product of linear representations

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