The Geometry of Generalised Spinʳ Spinors on Projective Spaces

In this paper, we adapt the characterisation of the spin representation via exterior forms to the generalised spinʳ context. We find new invariant spinʳ spinors on the projective spaces CPⁿ, HPⁿ, and the Cayley plane OP² for all their homogeneous realisations. Specifically, for each of these realisations, we provide a complete description of the space of invariant spinʳ for the minimum value of r for which this space is non-zero. Additionally, we demonstrate some geometric implications of the existence of special spinʳ spinors on these spaces.