Let us return to our case. We want to demonstrate that there are no excitations transforming in the adjoint of . So, we consider the special case of Frobenius reciprocity when is the adjoint rep of .

We have to go over all the tensor representations and verify if enters into using Eq. (5). We observe:

(the decomposition of as a rep of ) Now Eq. (5) reads:

[antisymm. tensors]

     [vectors]

The Type IIB SUGRA does not contain vectors, although does contain antisymmetric tensors: and .

But in the space of 2-forms on , the only subspace transforming in the adjoint of are where is parametrized by . But these are unphysical, a total gauge: .

This concludes the proof, that these are no states transforming in

The same conclusion can be reached by looking at the tables in the paper of H. J. Kim, L. J. Romans and P. van Nieuwenhuizen