Decomposing the Left Regular Representation Let’s take the left
regular representation of a finite group on its group algebra
and decompose it into irreducible representations. Our first
step is to compute the character of as a left -module. The nice
thing here is that it’s a permutation representation, and that
means we have a shortcut to calculating its character: is the
number of fixed point of the action of on the standard basis of
. That is, it counts the number of with . But this can only
happen if is the group identity, and in that case every element
is a fixed point. Thus we conclude where in the last line we use
the fact that evaluating the character of any representation at
the identity element gives the degree of that representation.
So, what does this tell us? Every irreducible representation
shows up in with a multiplicity equal to its degree. In
particular, it must show up at least once. That is, the left
regular representation contains all the irreducible
representations. Thus if are the irredu