I am working on a problem and I was very tempted to take for granted that for any complex character $\chi$ of a finite group $G$, the norm
$$\langle \chi, \chi \rangle = \frac{1}{|G|}\sum_{g\in G} \chi(g)\chi( g^{-1})$$
Is an integer. I have pondered this for awhile and I don't see a reason why it is necessarily true, but nontheless I have never seen any inner product of characters be anything but an integer.
My initial thought was that any character is the integral sum of irreducible characters, so the norm will be a sum of square integers, but that isn't sufficient
Am I missing something obvious here? If this is not true, can someone provide an explicit example for my repertoire of examples?
Best Answer
Let $G$ be finite, and restrict our attention to $G$-representations of finite degree over $\Bbb C$.
As you observe, to a $G$-representation $V$ we have a decomposition: $$V\cong V_1^{d_1}\oplus \dots \oplus V_h^{d_h},$$ where the $V_i$ are the irreducible representations corresponding to the irreduciblecharacters $\chi_i$ of $G$, i.e. the character $\chi$ of $V$ is $d_1\chi_1+\dots + d_h\chi_h$, and the orthogonality condition on the irreducible characters implies that $$(\chi|\chi) = \sum_{i,j=1}^h (d_i\chi_i|d_j\chi_j) = \sum_{i=1}^h (d_i\chi_i|d_i\chi_i)=\sum_{i=1}^h d_i^2(\chi_i|\chi_i) = \sum_{i=1}^h d_i^2,$$ which is, as you say, a sum of integers, and hence an integer. I am not sure why you say this is insufficient.