Descent algebras, hyperplane arrangements, and shuffling cards

Details Descent algebras, hyperplane arrangements, and shuffling cards Descent algebras, hyperplane arrangements, and shuffling cards

1998-01-20

arxiv.org/abs/math/9801089v4

Mathematics

Combinatorics

Scientific paper

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 $A$,$B$,$C$, $H_3$, 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.

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