While reading Polchinski's Chapter 1, I encountered the following on page 24,

"For example, the $(D-1)$ dimensional vector representation of $SO(D-1)$ breaks up into an invariant and a $(D-2)$-vector under the $SO(D-2)$ acting on the transverse directions, $$ \vec{v} = (v^1, 0, 0, \dots) + (0. v^2, v^2, \dots, v^{D-1}) $$

Thus, if a massive particle is in the vector representation of $SO(D-1)$, we will see a scalar and a vector when we look at the transformation properties under $S0(D-2)$. This idea extends to any representation : one can always reconstruct the full $SO(D-1)$ spin representation from the behaviour under $SO(D-2)$. "

I can show that the second excited states which is given by, $$ \alpha_{-1}^i \alpha_{-1}^j |0\rangle \bigoplus \alpha_{-2}^i |0\rangle $$ where $i,j$ runs from $\{2,D-1\}$ and treating them symmetric, the no. of excited states would be $\dfrac{(D+1)(D-2)}{2}$ which matches with the dimensions of a traceless symmetric irrep of $SO(D-1)$.

My question is how can we be sure by just matching numbers, and what does this Physically mean and is there a mechanism to do this consistently ? What does this business of "reconstructing $SO(D-1)$ representation from $SO(D-2)$" mean ?

I know a little bit about group theory like, Cartan Matrices, Dynkin Diagrams and Young Tableaux method for SU(N) theory, so it is fine if someone could give me a good reference. A precise answer would ofcourse be great :).

This post imported from StackExchange Physics at 2014-07-30 18:01 (UCT), posted by SE-user Jaswin