Character of direct sum of linear representations is sum of characters
From Groupprops
Statement
Suppose is a group,
is a field, and
are finite-dimensional linear representations of
over
. Denote by
the characters of the representations
respectively. Denote by
the direct sum of linear representations
and by
its character. Then, we have the following for any
:
Recall that the character of a linear representation is the function that sends any to the trace of the corresponding linear map.