Descent algebras, hyperplane arrangements, and shuffling cards

الأصل · EN

Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon's descent algebra and another using random walk on chambers of hyperplane arrangements. These coincide for types A,B,C, H₃, and rank two groups. Both notions have the same, simple eigenvalues. The hyperplane definition is especially natural and satisfies a positivity property when W is crystallographic and the relevant parameter is a good prime. The hyperplane viewpoint suggests interesting connections with Lie theory and leads to a notion of riffle shuffling for arbitrary real hyperplane arrangements and oriented matroids. Connections with Cellini's descent algebra are given.