# Deducing irreducible characters from non-irreducible characters

Say I have a complex character $$\alpha$$ of a finite group $$G$$ with the inner product $$(\alpha, \alpha) = 2$$. Since the only decomposition of $$2$$ as a sum of squares is $$2 = 1^2 + 1^2$$, the representation belonging two $$\alpha$$ must be a direct sum of two irreducible representations.

In the particular example I am working with, it is easy to calculate that for $$\chi_1$$ the trivial representation, the class function $$\gamma(g) = \alpha(g) - \chi_1(g)$$ has inner product $$(\gamma, \gamma) = 1$$. Is this enough to show that $$\alpha$$ is the direct sum of $$\chi_1$$ and $$\gamma$$? Since it's not even guaranteed that the representation for $$\alpha$$ has the subrepresentation for $$\chi_1$$, this does not seem like something I could deduce.

What about the case that $$(\gamma, \chi_1) = 0$$? Since this means that $$\gamma$$ is orthogonal to $$\chi_1$$ in the vector space of class functions on $$G$$, and all the irreducible characters are orthogonal to each other, does this imply $$\gamma$$ is a character (and thus irreducible)?

I would be interested in knowing this in the most generality possible, but I can supply concrete numbers if necessary, I just omitted them as the calculations are somewhat tedious.

Any difference of characters is still an integral linear combination of irreducible characters. So if your $$\gamma$$ has $$\langle \gamma, \gamma \rangle = 1$$, then $$\gamma$$ is $$\pm 1$$ times an irreducible character. In your case, it cannot be $$-1$$, so it is in fact an irreducible character.
• Could you elaborate on this? It sounds like you are using a general fact about orthonormal sets, but I am having trouble seeing how you get the implication $(\gamma, \gamma) = 1 \Rightarrow \gamma$ is $\pm 1$ an irreducible character. Jun 7, 2021 at 12:04
• @tolUene: the irreducible representations form an orthonormal set. So if $\gamma = \sum n_i \gamma_i$ then $\langle \gamma, \gamma \rangle = \sum n_i^2$. Note also that $n_i$ are integers. Jun 7, 2021 at 14:05