Benjamin Khademi: Cyclic sieving on closed walks in abelian Cayley graphs

Abstract

In this paper we study cyclic sieving on the set of closed walks of a particular length in abelian Cayley graphs. We interpret these walks as words in the alphabet of the generating set. We enumerate the number of such walks and their fixed point sets under the action of a cyclic group acting on the walks by way of cyclically shifting the letters of their corresponding words. We then show that this constitutes an instance of the cyclic sieving phenomenon. We show this first for cyclic graphs, then for circulant graphs before turning to the case of infinite rectangular grids. Finally, we also show it for Cayley graphs that are direct products of a finite number of circulant graphs.