# Representation of so(11) in terms of tensors

group-theorylie-algebrasrepresentation-theory

I'm going through some lecture notes I found online about Symmetry for Physicists and I got stuck in a problem of representations and tensors. The problem reads "The Lie algebra so(11) of rotations in 11 dimensions has an irreducible 55-dimensional representation.
Describe the representation module in terms of tensors." I've been thinking about it for a while but I'm not able to relate it to the simpler cases I found in the notes. Could someone provide me with a starting point?
Thank you

The defining representation generators of SO(n) are n×n antisymmetric real matrices $$M_{ij}$$, so, then, n(n-1)/2 of them, where i,j=1,...,n. Recall the generators you know from n=3.
For n=11, you then have 55 such matrices, the dimension of the so(11) algebra. The real structure constants of the commutators of these 55 matrices, $$M^a_{ij}$$, now indexed by a=1,....,55, among themselves$$^1$$, are the representation matrices of the adjoint representation, $$f^a_{bc}$$. They act on the space of 55-vectors, easy to construct from the rank-2 antisymmetric tensors $$M_{ij}$$.
1 $$[M_b,M_c]=f^a_{bc}M_a.$$