It is usually shown in the literature that massive light-cone gauge states for a closed bosonic string combine at mass level N=˜N=2 into representations of the little group SO(D−1) and then it is claimed (see, for example, David Tong's lecture notes on string theory) that it can be shown that ALL excited states N=˜N>2 fit into representations of SO(D−1). Is there a systematic way of showing this, as well as finding what those representations are? Maybe it was discussed in some papers, but I couldn't find anything remotely useful, everyone just seems to state this fact without giving a proof or a reference.
For OPEN strings at N=3 level the counting is:
(D−2)+(D−2)2+(D2)=(D−1)(D−2)2+((D+13)−(D−1)),
where on the LHS are
SO(24) representations, and on the RHS are
SO(25) representations. I'd like to find the same type of counting for CLOSED strings at any mass level
N,˜N>2, as claimed by Tong above.