Determinant Group Representations

If $G$ is a group and $(V, \rho)$ is a group representation of $G$ then we can obtain another representation of $G$ quite naturally.

Proposition 1: Let $G$ be a group and let $(V, \rho)$ be a group representation of $G$. Then $(\mathbb{C}^{\times}, \det_{\rho})$ where $\det_{\rho} : G \to C^{\times}$ is defined for all $g \in G$ by $\det_{\rho}(g) = \det (\rho(g))$ is a $1$-dimensional group representation of $G$.

• Proof: All that needs to be shown is that $\det_{\rho} : G \to C^{\times}$ is a group homomorphism. But this is simple. If $g_1, g_2 \in G$ then:

(1)

\begin{align} \quad \det_{\rho}(g_1g_2) = \det(\rho(g_1g_2)) = \det(\rho(g_1) \rho(g_2)) = \det(\rho(g_1)) \cdot \det (\rho(g_2)) = \det_{\rho}(g_1) \cdot \det_{\rho}(g_2) \end{align}

• Where the second last equality comes from the fact that the determinant function $\det$ is multiplicative. $\blacksquare$