Computing character tables and Cartan matrices of finite monoids with fixed point counting

In this paper we present an algorithm for efficiently counting fixed points in a finite monoid $M$ under a conjugacy-like action. We then prove a formula for the character table of $M$ in terms of fixed points and radical, which allows for the effective computation of the character table of $M$ over a field of null characteristic, as well as its Cartan matrix, using a formula from [Thi\'ery '12], again in terms of fixed points. We discuss the implementation details of the resulting algorithms and provide benchmarks of their performances.