Why is this character expression an integer?

Question

Let $\gamma$ be an $n$-dimensional complex representation of a finite group $G$ with character $\chi$ and let $e=c_0, c_1, ..., c_{\ell}$ be a set of conjugacy class representatives for $G$. In the case where $\gamma$ is faithful, I recently obtained the formula $$ \frac{1}{|G|} \prod_{i=1}^{\ell} (n-\chi(c_i))$$ as the size of a certain abelian group associated with $\gamma$ (see Theorem 3 in this paper if interested). In particular, this implies that this quantity is always an integer (if $\gamma$ is not faithful, then the product is 0).

Is there a nice way to see that this is an integer just using basic character theory?

Answer 1

Yes, this is an observation of H. Blichfeldt ( I think J-P. Serre also attributes it to Minkowski), which has been rediscovered many times over the years. Letting $1$ denote the trivial character, note that $\theta = \prod_{i = 1}^{\ell} ( \chi - \chi(c_{i})1) $ is an algebraic integer combination of characters of $G$, and also vanishes everywhere on $G$ except the identity, where its value is $\prod_{i = 1}^{\ell} ( n - \chi(c_{i})) .$ Hence $\theta$ is an algebraic integer multiple of the regular character of $G$, and $\langle \theta,1 \rangle = \frac{1}{|G|}\prod_{i = 1}^{\ell} ( n - \chi(c_{i})) $ is an algebraic integer ( which is easily seen to be rational by Galois theory, hence is a rational integer).

(Edit: In fact, it is only necessary to take the distinct values of $\chi(c_{i})$ without repetition in the product which is what Blichfeldt did).