# What is known about the projective representations of $\mathrm{SO}(n_1,n_2)$?

He${}$llo MO.

Let $\mathrm{O}(n_1,n_2)$ be the pseudo-orthogonal group. I am interested in its (continuous, not necessarily unitary, finite-dimensional) irreducible projective representations, for arbitrary $n_1,n_2$ (or, at least, the special cases $n_1=0,1$). I am having a hard time to find good characterisations thereof, or even a complete classification (some particular representations are discussed in some physics books).

I'd like to know how much is known about these representations. To keep the post focused, I will write some questions below, but in principle anything that is related to the problem at hand will be appreciated.

Some questions that come to mind:

1. Do all projective representations split into two classes, the regular and the "spinorial" ones (the former being those that map $2\pi\mapsto1$ and the latter those that map $2\pi\mapsto-1$)? In other words, are the only non-trivial two-cocycles those corresponding to $\pm1$?

2. May we have non-trivial central extensions? or are all the projective phases of topological origin (i.e., coming from the fact that $\mathrm{O}(n_1,n_2)$ is in general not simply-connected)?

3. How many labels do we need to specify a particular representation? are they restricted to half-integers (as is the case for $\mathrm{SO}(3)$ and $\mathrm{SO}(1,3)$)? what is the dimension of the corresponding representation, as a function of these labels?

4. Is there a one-to-one correspondence between irreducible projective representations and Young diagrams? or are the latter reserved to regular representations only?

5. How is an arbitrary tensor product of irreducible representations decomposed into direct sums of irreducible representations? (this is what we physicists call the Clebsch-Gordan decomposition).

6. Etc. Any information about the projective representations of $\mathrm{SO}(n_1,n_2)$ is welcome.

Thanks!

• FWIW: I think I know the answer to some of these questions. I am writing them anyway for two reasons: for completeness (as this may help future readers), and because I could use some confirmation (after all, I am a physicist :-P and I couldn't find an explicit answer anywhere). – AccidentalFourierTransform Sep 26 '17 at 20:52
• Could you edit in what you mean by representation? I guess continuous, not necessarily unitary, maybe finite dimensional? Bargmann (1954, §6e) answers your 1 and 2 for the identity component $\mathrm{SO}(p,m−p)^{\mathrm o}$ which he denotes $G'^m_p$, $m\geqslant3$. Briefly, central extensions (resp. projective representations) become trivial (resp. linear) when pulled back to the universal covering. Nowadays this is regarded as a consequence of semisimplicity and the Whitehead lemmas. – Francois Ziegler Sep 26 '17 at 21:54
• @FrancoisZiegler you're right. I fixed it. On a quick and superficial read, the reference looks really nice. Thank you! – AccidentalFourierTransform Sep 26 '17 at 22:08