Spinors and the quaternionic Poincar\'e group

When four dimensional spacetime R is considered as locally embedded on a larger manifold M, labelled by higher division algebra coordinates, a natural question to ask is how much of the symmetry properties of the larger space are inherited by R. Here this question is studied when M is a quaternion manifold. Of particular relevance is the absence of spinors in the linear representations of the symmetry group of the larger manifold and the emergence of new quantum numbers when, by Whitney sums, spinors are implemented on the vector bundles associated to the coset manifolds of the symmetry groups of M. A possible relation to the structures of the standard model is briefly discussed.