The Geometry of Generalised Spin${}^r$ Spinors on Projective Spaces

In this paper, we adapt the characterisation of the spin representation via exterior forms to the generalised spin${}^r$ context. We find new invariant spin${}^r$ spinors on the projective spaces ${\mathbb{C}\mathbb{P}}^n$, ${\mathbb{H}\mathbb{P}}^n$, and the Cayley plane ${\mathbb{O}\mathbb{P}}^2$ for all their homogeneous realisations. Specifically, for each of these realisations, we provide a complete description of the space of invariant spin${}^r$ spinors for the minimum value of $r$ for which this space is non-zero. Additionally, we demonstrate some geometric implications of the existence of special spin${}^r$ spinors on these spaces.