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Let G be a group, and $\rho : G \to GL(n, \mathbb{C})$ be a representation of $G$. Then we also get the conjugate representation $\rho^* : G \to GL(n, \mathbb{C})$, where $\rho^*(g) = \overline{\rho(g)}$. This definition is naïve, but it will suffice here.

We say that $\rho$ is a real representation if there exists some $M \in GL(n,\mathbb{C})$ such that $M\rho(g)M^{-1} \in GL(n,\mathbb{R}) ~\forall~ g \in G$. If $\rho$ is not real, but still $\rho \cong \rho^*$, we say that $\rho$ is pseudoreal.

I believe it is a fact (though I don't have a reference) that the tensor product of two pseudoreal representations is real, but I've never quite understood why. Is there an elementary proof, and/or an easy way to understand this?

Simple examples: The fundamental representation $\mathbf{2}$ of $SU(2)$ is pseudoreal, but $\mathbf{2}\otimes\mathbf{2} = \mathbf{3}\oplus\mathbf{1}$ is real (the $\mathbf{3}$ is the fundamental rep. of $SO(3)$, lifted to $SU(2) \cong \text{Spin}(3)$). Similarly, if we consider $SU(2)\times SU(2)$, then $(\mathbf{2}, \mathbf{2}) = (\mathbf{2},\mathbf{1})\otimes(\mathbf{1},\mathbf{2})$ is real, being the lift to $SU(2)\times SU(2) \cong \text{Spin}(4)$ of the fundamental rep. of $SO(4)$.

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up vote 3 down vote accepted

I can at least tell you what happens in the case when $G$ is finite. I would have to think a bit about how this extends to compact Lie groups, but I would think that it should extend quite well.

What you are describing is a special case of the theory of Schur indices. Let the complex representation $\rho$ have character $\chi$. The basic question one asks is:

if you adjoin to $\mathbb{Q}$ all values of $\chi$, and denote the resulting field by $\mathbb{Q}(\chi)$, then can $\rho$ be realised by matrices with coefficients in $\mathbb{Q}(\chi)$?

The answer is, as you have observed, "not necessarily", and the smallest degree of an extension $F/\mathbb{Q}(\chi)$ such that $\rho$ is realisable over $F$ is called the Schur index of $\chi$. This is also the smallest integer $m$ such that $\rho^{\otimes m}$ is realisable over $\mathbb{Q}(\chi)$. You are describing the particular case of so-called symplectic representations, which have Schur index 2. All this is very nicely treated in Curtis and Reiner, Methods of Representation Theory, with applications to finite groups and orders, Chapter 9, sections 73, 74. See in particular Corollary (74.8).

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Thanks Alex, that's useful. I can't immediately see how it answers my question (perhaps I'm being dense...), but I'll check out the reference. – Rhys Dec 7 '12 at 12:57
Your question is about the special case of Schur index 2: $\rho\cong \rho^*$ is equivalent to the character of $\rho$ being real valued. But if at the same time $\rho$ is not realisable over $\mathbb{R}$, then $\rho$ is called symplectic, or sometimes quaternionic. Such beasts have Schur index two, and the answer to your question is therefore given by Corollary (74.8) in Curtis and Reiner. – Alex B. Dec 7 '12 at 13:02

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