Twisted Flato-Fronsdal theorem for higher-spin algebras

We explore the relation between the singleton and adjoint modules of higher-spin algebras via $so(2,d)$ characters. In order to relate the tensor product of the singleton and its dual to the adjoint module, we consider a heuristic formula involving symmetrization over the variables of the character. We show that our formula reproduces correctly the adjoint-module character for type-A (and its high-order extensions) and type-B higher-spin gravity theories in any dimension. Implications and subtleties of this symmetrization prescription in other models are discussed.