Cellini's Descent Algebra and Semisimple Conjugacy Classes of Finite Groups of Lie Type

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. We conjecture that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini which uses the affine Weyl group. This conjecture is confirmed in special cases such as type C odd characteristic and the identity conjugacy class in type A. Models of card shuffling, old and new, arise naturally. Type A shuffles lead to interesting number theory involving Ramanujan sums. It is shown that a proof of our conjecture in type C even characteristic would give an alternate solution to a problem in dynamical systems. An idea is offered for how, at least in type A, to associate to a semisimple conjugacy class an element of the Weyl group, refining the map to conjugacy classes. This is confirmed for the simplest nontrivial example.