Kinematics and dynamics of quantum walks in terms of systems of imprimitivity

We build systems of imprimitivity (SI) in the context of quantum walks and provide geometric constructions for their configuration space. We consider three systems, an evolution of unitaries from the group SO ₃ on a low dimensional de Sitter space where the walk happens on the dual of SO ₃, a standard quantum walk whose SI live on the orbits of stabilizer subgroups (little groups) of semidirect products describing the symmetries of 1  +  1 spacetime, and automorphisms (walks are specific automorphisms) on distance-transitive graphs as an application of the constructions. (paper)