Character of direct sum of linear representations is sum of characters

Statement

Suppose $G$ is a group, $K$ is a field, and $\rho_1:G \to GL(V_1), \rho_2:G \to GL(V_2)$ are finite-dimensional linear representations of $G$ over $K$. Denote by $\chi_{\rho_1}, \chi_{\rho_2}$ the characters of the representations $\rho_1,\rho_2$ respectively. Denote by $\rho_1 \oplus \rho_2$ the direct sum of linear representations $\rho_1,\rho_2$ and by $\chi_{\rho_1 \oplus \rho_2}$ its character. Then, we have the following for any $g \in G$:

$\chi_{\rho_1 \oplus \rho_2} = \chi_{\rho_1}(g) + \chi_{\rho_2}(g)$

Recall that the character of a linear representation is the function that sends any $g$ to the trace of the corresponding linear map.

Related facts

• Character of tensor product of linear representations is product of characters