Coxeter Groups and representations of $ + $ type Background:
A Coxeter group is generated by reflections, which all have the real eigenvalues $ \pm 1 $, so intuitively it makes sense that all the representations would be real. Certainly every Coxeter group must have one faithful real valued irrep, since every finite Coxeter group is just a Euclidean reflection group.
Question:
I have noticed that all the irreps of a finite Coxeter group are of $ + $ type (I have not checked every finite Coxeter group but this seems to be the pattern) (by $ + $ type I mean they have Frobenius-Schur indicator $ +1 $). I would imagine this fact (if true) is well know. Does anyone know an explanation/proof/reference for this?
Follow Up Question:
More generally, does anyone know any necessary or sufficient (or necessary and sufficient!) conditions for a finite group to have all $ + $ type irreps?
 A: Some stuff I scrounged up. First, an easier result: an element $g$ of a finite group $G$ is real if it is conjugate to $g^{-1}$, or equivalently if the character of $g$ is always real in every representation. So the following conditions are equivalent:

*

*Every element of $G$ is conjugate to its inverse.

*Every character of $G$ is real.

*Every finite-dimensional representation of $G$ is either real or quaternionic.

Such groups are called ambivalent. This is a necessary condition for all representations being real; the symmetric groups obviously satisfy it, but I'm not sure if all finite Coxeter groups do (edit: they do, see the comments). However, it turns out to be satisfied by all Weyl groups.
The stronger condition that every representation is real is called being totally orthogonal; apparently it's known (see the link) that any such group must be generated by involutions. It doesn't seem like necessary and sufficient conditions are known in general.
Finally, the Weyl groups satisfy an even stronger condition which implies total orthogonality, that every representation is realizable over $\mathbb{Q}$: this is claimed in Humphreys in Section 8.10 and the citation is to Benard's On the Schur indices of characters of the exceptional Weyl groups. Such a group must in particular satisfy the property that every character is integer-valued and this condition is called being rational.
