Nilpotent representations over equioriented cyclic quivers

Abstract

We give a geometric description of nilpotent representations of equioriented cyclic quivers over a field via an explicit bijection. Over a finite field, this yields formulas for the number of nilpotent representations and for the probability that a representation is nilpotent. We also give a refinement by rank. Over the Boolean semiring, we identify nilpotent semirepresentations with directed acyclic graphs, derive a recursive formula for their number, and determine the asymptotic decay rate of the probability of nilpotence.