Descent algebras, hyperplane arrangements, and shuffling cards
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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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- 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: email@example.com, firstname.lastname@example.org
- 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