Descent algebras, hyperplane arrangements, and shuffling cards

Author:
Jason Fulman

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

Published electronically:
October 19, 2000

MathSciNet review:
1625753

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 ,,,, 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 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

Email:
fulman@dartmouth.edu, fulman@math.stanford.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05055-3

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