Descent algebras, hyperplane arrangements, and shuffling cards

1998/01/20 by Jason Fulman, Fulman, Jason
Mathematics · #20F55 #20G40 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:20F55 #msc:20G40

paper · pdf · doi:10.48550/arxiv.math/9801089

openalex publication_date 1998/01/20 · arxiv created 1999/07/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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