Finite-dimensional vertex in AdS

Now returning to the AdS point of view.

We will construct the vertex in , for some finite-dimensional .

More specifically, we take a family of representations parametrized by an integer :

    

where:

• Remember that is the symmetry algebra of , and the symmetry algebra of , see also here.

• Let us use greek indices for the fundamental of , and latin indices for the fundamental of , as we already did.

• is some specific Young symmetrizer, which acts as follows. The space consists of the tensors ; the operation first antisymmetrizes , and then symmetrizes and . Similarly the space is the space of tensors which are antisymmetrized and then symmetrized in the same way.