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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Descent algebras, hyperplane arrangements, and shuffling cards
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by Jason Fulman PDF
Proc. Amer. Math. Soc. 129 (2001), 965-973 Request permission

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 $A$,$B$,$C$,$H_3$, and rank two groups but not always. 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 deep connections with Lie theory and leads to a notion of riffle shuffling for arbitrary real hyperplane arrangements and oriented matroids.
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Additional Information
  • Jason Fulman
  • Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
  • Address at time of publication: Department of Mathematics, Stanford University, Stanford, California 94305
  • MR Author ID: 332245
  • Email: fulman@dartmouth.edu, fulman@math.stanford.edu
  • Received by editor(s): January 30, 1998
  • Received by editor(s) in revised form: May 18, 1998, and July 15, 1999
  • Published electronically: October 19, 2000
  • Communicated by: John R. Stembridge
  • © Copyright 2000 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 965-973
  • MSC (1991): Primary 20F55, 20P05
  • DOI: https://doi.org/10.1090/S0002-9939-00-05055-3
  • MathSciNet review: 1625753