# How to check whether a representation $G\to\mathrm{GL}(n,\Bbb R)$ is irreducible?

I know there is a very beautiful theory for representations over $$\Bbb C$$, especially the character theory makes it almost trivial to check whether a given representation $$G\to\mathrm{GL}(n,\Bbb C)$$ is irreducible.

But how can I check this in a similarly algorithmic fashion for representations over $$\Bbb R$$? I am specifically interested in the case of finite groups.

Question: Given a finite group $$G$$ and a representation $$\rho:G\to\mathrm{GL}(n,\Bbb R)$$. How to determine (algorithmically) whether $$\rho$$ is irreducible?

Note I

I am aware of Frobenius-Schur indicator but I cannot understand whether and how it helps me for my question. At first, I do not have a representation over $$\Bbb C$$ to start with. And I am not really interested in transforming my irreducible representation over $$\Bbb R$$ into one or more irreducible representation over $$\Bbb C$$.

Note II

I avoid using the term "real representation" as it seems to have the meaning of a representation over $$\Bbb C$$ with a real valued character. I am not very familiar with the connection of this term and "representations over $$\Bbb R$$" that I use. But please enlighten me.

• Possible duplicate of math.stackexchange.com/questions/818543/… – user10354138 Oct 1 '18 at 14:50
• When a real irrep $V$ is complexified, precisely three things can happen, and these are classified by the Frobenius-Schur indicator. – Joppy Oct 1 '18 at 14:54
• @Joppy I have a hard time grasping how these indicators work in "my direction". What I see is: they can classify irrep. representations over $\Bbb C$ into three classes. But what do they tell me when I start from a representation over $\Bbb R$ and what do they tell me about irreducibility? – M. Winter Oct 1 '18 at 14:57
• @user10354138 The other question seems to be about deciding whether a given complex irreducible is equivalent to a real representation. Here we are given a real representation as input. Clearly if it is irreducible over ${\mathbb C}$ then it is irreducible over ${\mathbb R}$, and if it is the sum of more than two complex irreducibles then it is reducible over ${\mathbb R}$. The tricky case is where it is the sum of two complex irreducibles. In that case you would need to idenify them. – Derek Holt Oct 1 '18 at 15:01

One of the links in the comments below the question brought me to the following website, which contains a nice characterization of irreducible representations over $\Bbb R$:
A representation $\rho:G\to\mathrm{GL}(n,\Bbb R)$ with character $\chi=\mathrm{tr}(\rho)$ is irreducible, if and only if $$\|\chi\|^2+\nu(\chi)=2.$$
Here $\|\chi\|^2$ is the squared norm of the character, and $\nu$ is the Frobenius-Schur indicator defined by
$$\|\chi\|^2=\frac1{|G|}\sum_{g\in G}\chi^2(g),\qquad \nu(\chi):=\frac1{|G|}\sum_{g\in G}\chi(g^2).$$