  Using projective spaces as examples of toric manifolds, we examine
K-theoretic fixed point localization. On the one hand, we will see how the
permutation-equivariant theory of the point target space emerges as a necessary
ingredient. On the other hand, we will completely characterize the genus-0
permutation-equivariant quantum K-theory of the given toric manifold in terms
of such theory for the point, and a certain recursion relation.
