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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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