Special spinors play a key role in understanding geometric and topological properties of manifolds. This thesis investigates how symmetry helps construct and classify special spinors on homogeneous spaces, both in classical and generalised spinʳ settings. A central idea is that symmetry provides algebraic control over spinorial equations, making explicit constructions and classification results possible.


In the first part of the thesis, we establish new results in the classical spin setting. First, we provide a combinatorial description of invariant spinors on flag manifolds in terms of the root system of the corresponding Lie algebras. We then give the first complete classification of invariant generalised Killing spinors on three-dimensional Lie groups equipped with arbitrary left-invariant metrics. Remarkably, their existence and the symmetry properties of the associated endomorphism depend only on the underlying Lie algebra, and not on the left-invariant metric itself. Finally, we construct the first known examples of invariant generalised Killing spinors whose associated endomorphisms have more than four distinct eigenvalues.

The second part of the thesis concerns a generalisation of spin structures known as spinʳ structures. This framework allows spinorial techniques to be applied to all oriented manifolds, even if they are not spin. We systematically develop the theory of invariant spinʳ structures on homogeneous spaces and provide a representation-theoretic classification. We then compute the minimal r needed for each homogeneous realisation of spheres to admit invariant spinʳ structures, and establish an equivalence between certain holonomy lifts and the existence of special invariant spin structures on spheres. Finally, we construct explicit special spinʳ spinors on complex and quaternionic projective spaces that encode geometric information about these manifolds.

Bringing together tools from algebra, topology, geometry, and representation theory, this thesis advances the study of classical and spinʳ spinors, and opens new directions for exploring geometric structures through spinorial techniques.