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