# Can one reformulate tensor methods and young tableaux to account for spinor representations on $\operatorname{SO}(n)$?

Standard tensor methods and Young tableaux methods don't give you the spinor reps of $\operatorname{SO}(n)$.

1. Is this because spinor representation are projective representations? If so, where does this caveat of projective representations enter this formulation of finding irreducible representations?

2. Given that 'standard' tensor methods and Young tableaux (i.e. ones that you might find in a physics book on lie algebras) don't give you spinor reps, are there generalized Young tableaux methods that give you spinor reps?

Edit: Just to give you an idea where I am coming from, I am a physicist, so I am sort of asking for the dummies guide to enumerating all possible representations without missing any.

• 1) Yes. Tensoring starting from ordinary representations will only get you ordinary representations. 2) I think so, but I'm not familiar with them. – Qiaochu Yuan May 31 '13 at 21:24

## 1 Answer

Analogues of Young tableaux exist for all semisimple Lie groups/algebras, even for Kac-Moody algebras. (Some look like minor modifications of YT, other models are more geometric, called "path models".) They can be used to work with spin representations of orthogonal groups as a very special case. You can find many papers on this on Peter Littelmann's web page.

For instance:

"The path model of representations", Proceedings of the ICM Zürich 1994, Birkhäuser Verlag, Basel--Boston, (1995), pp. 298--308.