Irreducible representations of $SO(N)$

Question

I'm looking for a reference/proof where I can understand the irreps of $SO(N)$. I'm particularly interested in the case when $N=2M$ is even, and I'm really only interested in the reps that come from taking exterior powers of $\mathbb{R}^{2M}$. Doing some numerical experiments, it seems like for general $M$ the following facts are true:

• $\bigwedge^m \mathbb{R}^{2M}$ is a real irrep for $m\neq M$.
• $\bigwedge^m \mathbb{R}^{2M} \simeq \bigwedge^{2M-m} \mathbb{R}^{2M}$ for $m\neq M$.
• $\bigwedge^M \mathbb{R}^{2M}$ decomposes into two irreps of the same dimension. These irreps are real when $M$ is even and complex-conjugates of each other when $M$ is odd.

Does anyone know how I can prove this? I think proving $\bigwedge^m \mathbb{R}^{2M} \simeq \bigwedge^{2M-m} \mathbb{R}^{2M}$ seems straightforward, but I am not sure how to prove irreducibility for any of these cases.

Answer 1

You may also take a look at "Representation Theory of Semisimple Groups" by Knapp. The details are given in Chapter IV, §5, Examples 3 and 4.

Answer 2

Okay, I found a reference: this is worked out (over $\mathbb{C}$ but the result over $\mathbb{C}$ should imply the result over $\mathbb{R}$) as Theorem 19.2 in Fulton and Harris' Representation Theory: A First Course. Fulton and Harris do it by restricting to a copy of the symplectic group which is not what I expected.