An irreducible representation $\rho$ (with character $\chi$) of a finite group is called a "real" representation if its Frobenius-Schur indicator is 1: $$\frac{1}{\lvert G \rvert} \sum_{g \in G} \chi\left(g^2\right) = 1$$ Alternatively, it is real if it has a symmetric bilinear, $G$-equivariant form.

Finite groups can have real, quaternionic and complex representations. Some groups like the symmetric group have only real representations.

Which groups have only real representations?

  • $\begingroup$ @ladisch, yes, thank you! $\endgroup$ – Turion Jun 16 '16 at 8:59
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    $\begingroup$ You probably know that all irreducible characters are real-valued iff every group element is conjugate to its inverse. But this means just that all irreps are real or quaternionic. I don't know a characterization of groups with only irreps of real type. Maybe related: mathoverflow.net/q/109045/10266 $\endgroup$ – ladisch Jun 16 '16 at 9:21
  • $\begingroup$ @ladisch, I didn't know that, but that's also valuable. That would mean that in such a group, every representation is self-dual, right? $\endgroup$ – Turion Jun 16 '16 at 9:32
  • $\begingroup$ Yes, then every representation is self-dual. $\endgroup$ – ladisch Jun 16 '16 at 9:36
  • $\begingroup$ Thanks for linking the question. An article linked therein says that at least in the 70's, this was not so clear. Another linked article calls groups with my desired property "totally orthogonal", and apparently this is ongoing research. $\endgroup$ – Turion Jun 16 '16 at 9:50

The correct term for this kind of groups is called "totally orthogonal" (e.g. orthogonal groups have this property). It was coined in this article by Wang and Grove, together with many propositions about their structure. For example, it contains:

Theorem 3.3 If $G$ is totally orthogonal, it is generated by involutions.


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