I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point.
Proctor describes this rule for $m = 2$. In order to do this, he actually works with the disconnected group $O(n)$ instead of $SO(n)$. The representation theory of $O(n)$ is a tiny bit more complicated than the one of $SO(n)$, but turns out to be natural in this setting as it leads to simpler branching rules in the end. Now Proctor also deals with non-tensor representations of $\mathfrak{so}(n)$, i.e. representations of $\operatorname{Spin}(n)$ (which he calls $\tilde{SO}(n)$) which do not descend to $SO(n)$. To deal with those, he also passes to the corresponding two-component group, that he calls $\tilde{O}(n)$.
Now here is the situation with the representation theory of the various groups involved:
For $SO(n)$, the theory is straightforward and very well-known. Irreps are in bijection with Young diagrams of height at most $\lfloor \frac{n}{2} \rfloor$ (the rank of the group).
For $O(n)$, there is also a beautiful theory, essentially due to Weyl, and expounded clearly for example in King and Welsh. Irreps are in bijection with Young diagrams whose first column is allowed to be taller than than the rank, all the way to $n$. There are thus roughly two irreps of $O(n)$ for each irrep of $SO(n)$, obtained by "reflecting" the bottom of the first column through the horizontal line at height $\frac{n}{2}$. This correspondance is exact for odd $n$ (and indeed we then have $O(n) \simeq SO(n) \times \mathbb{Z}/2\mathbb{Z}$), and involves some fine print for even $n$.
For $\operatorname{Spin}(n)$, the representation theory is the same as for the Lie algebra $\mathfrak{so}(n)$, so is also straightforward (highest-weight theory). Irreps are in bijection with Young diagrams of height at most $\lfloor \frac{n}{2} \rfloor$ that are additionally allowed to have an extra "half-column" at the beginning.
Now for the Pin groups, things get really tricky. Proctor cheerfully says "A two-fold cover of the orthogonal group $O_N$ is denoted $\tilde{O}_N$", which already brings the first difficulty: at least two different covers of this sort are commonly considered, usually denoted by $\operatorname{Pin}^+(n)$ and $\operatorname{Pin}^-(n)$; and I have no idea which one of them Proctor has in mind.
Proctor then presents a parametrization of the irreps of this "$\tilde{O}_N$" (last paragraph of p.305 and first paragraph of p.306). However, he does not give any proof nor reference for this parametrization.
I think it is eventually humanly doable to figure that out on my own. But maybe someone already has a good understanding of this? If so, would you help me answer the following questions?
Is Proctor's description correct for $\operatorname{Pin}^+$? For $\operatorname{Pin}^-$? For both?
Is this written up somewhere?
