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Descent algebras, hyperplane arrangements, and shuffling cards

Author: Jason Fulman
Journal: Proc. Amer. Math. Soc. 129 (2001), 965-973
MSC (1991): Primary 20F55, 20P05
Published electronically: October 19, 2000
MathSciNet review: 1625753
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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

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
Article copyright: © Copyright 2000 American Mathematical Society