Let $\chi_{R}$ denote the character of the right regular representation $R$. Compute the inner product $(\chi_{R},\chi_{R})$ directly and also by decomposing $R$ into a sum of irreducible representations.
But I have the following difficulties?
1- what is the character of the right regular representation? I searched my book (Ernest Vinberg "Linear representations of groups ") but I did not find the definition.
2-How can I decompose $R$ into a sum of irreducible representations? and I think the value of the inner product should be the same .... correct?
EDIT :
I got the answer for my first question, it is 0 if $g \neq e$ and its n (order of the group) if $g = e$.
Still I am working on the second question. but I appreciate any help on it.
EDIT 2: I have known that I will use that $\chi_{R} = \sum_{i = 1}^{q} d_{\pi_{i}} \chi_{\pi_{i}} $ but still I do not know how to reach the results I obtained in the first part, could anyone help me please?