Abstract
We describe a cyclic action of order q -1 on the classical space of generalized parking functions for Weyl groups W when q is prime, as well as a similar action when q is a power of a prime on an isomorphic "semi-classical" parking space. We show that these actions agree with the action described by Armstrong, Reiner, and Rhoades in their parking conjectures.
Along the way, we also conjecture minimal rings in which Weyl group elements have a parametrized Smith normal form.