Let $G$ be a finite group. For any $\mathbb{C}G$-module $W$ we define its $\textbf{Frobenius-Schur -ndicator}$ to be $$ \iota_W=\frac{1}{|G|}\sum_{g\in G}\chi_W(g^2), $$ where $\chi_W$ is the character of $W$.

My question is whether the following statement is true:


If $U$ is an irreducible $\mathbb{R}G$-module, then its character has one of the following forms:

  1. $\chi_U$, and $U$ is an irreducible $\mathbb{C}G$-module,
  2. $2\operatorname{Re}\chi_W$, for some irreducible $\mathbb{C}G$-module $W$.

Moreover, for any irreducible $\mathbb{C}G$ module $V$, one of the following is true:

  1. $\iota(V)=1$, then $V$ is an irreducible $\mathbb{R}G$-module,
  2. $\iota(V)\neq 1$, then $2\operatorname{Re}\chi_V$ is a character of some irreducible $\mathbb{R}G$-module.

We can define the inner product

$$\langle \chi_1, \chi_2 \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{\chi_1}(g) \chi_2(g)$$

of characters for representations over $\mathbb{R}$ in the same way as over $\mathbb{C}$ (of course here we have $\overline{\chi} = \chi$). It is still true that

$$\langle \chi_V, \chi_W \rangle = \dim \text{Hom}_G(V, W)$$

but the computation of $\text{Hom}_G(V, W)$ is slightly more complicated. We still have that nonisomorphic irreducibles are orthogonal, but if $V$ is irreducible then

$$\langle \chi_V, \chi_V \rangle = \dim \text{End}_G(V)$$

can take one of three possible values, corresponding to the three finite-dimensional real division algebras over $\mathbb{R}$ (Schur's lemma guarantees that $\text{End}_G(V)$ is a division algebra):

  1. If $\text{End}_G(V) \cong \mathbb{R}$ then $\langle \chi_V, \chi_V \rangle = 1$.
  2. If $\text{End}_G(V) \cong \mathbb{C}$ then $\langle \chi_V, \chi_V \rangle = 2$.
  3. If $\text{End}_G(V) \cong \mathbb{H}$ then $\langle \chi_V, \chi_V \rangle = 4$.

Now, if $V$ is an irreducible real representation, consider the complexification $V \otimes \mathbb{C}$. It has the same character as $V$, and hence $\langle \chi_V, \chi_V \rangle$ takes the same values.

  1. If $\langle \chi_V, \chi_V \rangle = 1$ then $V \otimes \mathbb{C}$ is irreducible.
  2. If $\langle \chi_V, \chi_V \rangle = 2$ then $V \otimes \mathbb{C}$ is the direct sum of two nonisomorphic irreducibles, whose characters must be complex conjugates of each other. These are given by $V$ together with the action of $\text{End}_G(V) \cong \mathbb{C}$, where there are two ways of identifying $\text{End}_G(V)$ with $\mathbb{C}$ related by complex conjugation.
  3. If $\langle \chi_V, \chi_V \rangle = 4$ then $V \otimes \mathbb{C}$ is the direct sum of two isomorphic irreducibles, whose character must be real. These are given by $V$ together with the action of any copy of $\mathbb{C}$ inside $\text{End}_G(V) \cong \mathbb{H}$.

(This does not quite follow from a dimension count but it does follow from the identity $\text{End}_G(V \otimes \mathbb{C}) \cong \text{End}_G(V) \otimes \mathbb{C}$.)

This gives your first statement. The second statement is also true (each of the three cases above corresponds to a possible value of the Frobenius-Schur indicator) but the proof escapes me for the moment.


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