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.

  • $\begingroup$ Possible duplicate of math.stackexchange.com/questions/818543/… $\endgroup$ – user10354138 Oct 1 '18 at 14:50
  • $\begingroup$ When a real irrep $V$ is complexified, precisely three things can happen, and these are classified by the Frobenius-Schur indicator. $\endgroup$ – Joppy Oct 1 '18 at 14:54
  • $\begingroup$ @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? $\endgroup$ – M. Winter Oct 1 '18 at 14:57
  • 1
    $\begingroup$ @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. $\endgroup$ – 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

$$\|\chi\|^2=\frac1{|G|}\sum_{g\in G}\chi^2(g),\qquad \nu(\chi):=\frac1{|G|}\sum_{g\in G}\chi(g^2).$$


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