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

By algebraic group theory, there is a map from the semisimple conjugacy classes of a nite 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. This conjecture is con rmed for type C in odd characteristic, and for type A2. Connections with old and new card shu ing models are indicated. An idea is o ered for how, at least in type A, to associate to a semisimple conjugacy class an element of the Weyl group, re ning the map to conjugacy classes. This is con rmed for the simplest nontrivial example.