Descent algebras, hyperplane arrangements, and shuffling cards
Author:
Jason Fulman
Journal:
Proc. Amer. Math. Soc. 129 (2001), 965973
MSC (1991):
Primary 20F55, 20P05
Published electronically:
October 19, 2000
MathSciNet review:
1625753
Fulltext PDF Free Access
Abstract 
References 
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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 ,,,, 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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 Bidigare, P., Hyperplane arrangement face algebras and their associated Markov chains, Ph.D. Thesis, University of Michigan, 1997.
 [BHR]
 Bidigare, P., Hanlon, P., and Rockmore, D., A combinatorial description of the spectrum of the Tsetlin library and its generalization to hyperplane arrangements, Duke Math. J. 99 (1999), 135174. CMP 99:16
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 Brown, K. and Diaconis, P., Random walk and hyperplane arrangements. Ann. of Probab. 26 (1998), 18131854. CMP 99:09
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 Carter, R., Finite groups of Lie type. John Wiley and Sons, 1985. MR 87d:20060
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 Carter, R., Conjugacy classes in the Weyl group. Composito Math. 25 (1972), 159. MR 47:6884
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 Fulman, J., Semisimple orbits of Lie algebras and card shuffling measures on Coxeter groups, J. Algebra 224 (2000), 151165.
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 Fulman, J., Counting semisimple orbits of finite Lie algebras by genus, J. Algebra 217 (1999), 170179. CMP 99:15
 [F3]
 Fulman, J., The combinatorics of biased riffle shuffles. Ann. of Combin. 2 (1998), 16.
 [F4]
 Fulman, J., Cellini's descent algebra, dynamical systems, and semisimple conjugacy classes of finite groups of Lie type, http://xxx.lanl.gov/abs/math.NT/9909121.
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 Fulman, J., Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting, to appear in J. Algebra.
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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:
http://dx.doi.org/10.1090/S0002993900050553
PII:
S 00029939(00)050553
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
