A summary and \or reference to the theory of REAL representation theory

abstract-algebrafinite-groupsrepresentation-theory

In fulton and harris there is a short discussion of real representations which is unsatisfactory to me.

In $\mathbb{C}$ we have a great theory-

  1. We know how many irreducible representations there are
  2. We have a simple relation about their dimensions that in particular bounds the dimensions of irreducible nicely
  3. Most importantly, we have characters which are great tools to let us decompose a given representation to irreducible ones.

I want to understand what the analogs of those are in $\mathbb{R}$.

Attempt

My attempts give 'algorithms' to solve those questions, but I'd prefer formulas if that makes sense (like the number of conjugacy classes, etc).

1/2.

We can use Artin Wedderburn and the Frobenius theorem to know the group algebra breaks up to a sum of $M_{n\times n}(D)$ for $D$ one of $\mathbb{R},\mathbb{C},\mathbb{H}$. That gives us good bounds on the irreducible representations and I guess a 'formula' that instead of just their dimensions and amount, involves $Dim Hom_\mathbb{R}(V,V)$

3.

Suppose you found all complex representations.
The cheapest attempt is to say okay to understand a real representation $V$ I'll complexify it to $V'$. Then if $V = \oplus V_i$ then $V' = \oplus V_i '$, though the other direction is not true, so I have a technical way to decompose a representation assuming I know the complex ones, by going over all subsets of the summands of the decomposition of $V'$ and see if their sum comes from some $W \otimes \mathbb{C}$ for $W$ a subspace of $V$.
Fulton-Harris gives a criterion to when a represenation is real (complexified of real), but that isn't the same as checking for me if the sum comes from some $W \otimes \mathbb{C}$, so I don't even understand why this criterion is important.

What are more accurate\systematic things one can say or good references?

Best Answer

I enjoyed Joppy's sources, writing this answer to partially answer and close the question.

As Joppy mentioned https://math.mit.edu/~poonen/715/real_representations.pdf is a cool source.

Here is the short summary if I understand correctly-

Take the irreducible representations over $\mathbb{C}$, and pair them up by conjugation. Call a pair weird if $V = \bar{V}$. For nonweird pairs, there is a unique irreducible representation over $\mathbb{\mathbb{R}}$, $W$, with $W_{\mathbb{C}} = V \oplus \bar{V}$, and in particular the restriction to $\mathbb{R}$ of $V,\bar{V}; _{\mathbb{R}}V$ gives $W$.

For a weird pair, there are two options. First, there is the option $V$ is real, and so is $V=W_{\mathbb{C}}$, in this case $W$ is unique, irreducible, and can be obtained via finding a bilinear symmetric form and following the proof of FS.

In the second option, $V$ is quaternionic, in which case $V \oplus V = W_{\mathbb{C}} $ for a unique irreducible $W$, and $W$ is the restriction of $V$ to $_{\mathbb{R}}W$.

The above description gives a bijection between pairs of complex irreducibles and the real representations.

This also answers 2 as best as possible (you can relate the above cases of $W_\mathbb{C}$ with what $D(W)$ is).

For 3, again thanks to Joppy for the source. We have the frobenius schur indicator $v(\chi)$ associated to a complex representation.

Given a real representation $W$, if it's irreducible the notes explain how $W_\mathbb{C}$ can split. You can check in all cases $||\chi||^2 + v(\chi) = 2$ (this is a numerical miracle though that we have a condition by the character as far as I can see), and otherwise $||\chi||^2 + v(\chi) > 2$. Finally for two distinct real representations, again the explanation shows $\chi_1,\chi_2$ are orthogonal (since the complex reps are distinct).

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