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

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.

- Dave Bayer and Persi Diaconis,
*Trailing the dovetail shuffle to its lair*, Ann. Appl. Probab.**2**(1992), no. 2, 294–313. MR**1161056** - François Bergeron and Nantel Bergeron,
*Orthogonal idempotents in the descent algebra of $B_n$ and applications*, J. Pure Appl. Algebra**79**(1992), no. 2, 109–129. MR**1163285**, DOI https://doi.org/10.1016/0022-4049%2892%2990153-7 - F. Bergeron, N. Bergeron, R. B. Howlett, and D. E. Taylor,
*A decomposition of the descent algebra of a finite Coxeter group*, J. Algebraic Combin.**1**(1992), no. 1, 23–44. MR**1162640**, DOI https://doi.org/10.1023/A%3A1022481230120 - François Bergeron and Nantel Bergeron,
*Symbolic manipulation for the study of the descent algebra of finite Coxeter groups*, J. Symbolic Comput.**14**(1992), no. 2-3, 127–139. MR**1187228**, DOI https://doi.org/10.1016/0747-7171%2892%2990032-Y - Bidigare, P., Hyperplane arrangement face algebras and their associated Markov chains,
*Ph.D. Thesis*, University of Michigan, 1997. - 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), 135–174. - Brown, K. and Diaconis, P., Random walk and hyperplane arrangements.
*Ann. of Probab.***26**(1998), 1813-1854. - Roger W. Carter,
*Finite groups of Lie type*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR**794307** - R. W. Carter,
*Conjugacy classes in the Weyl group*, Compositio Math.**25**(1972), 1–59. MR**318337** - Fulman, J., Semisimple orbits of Lie algebras and card shuffling measures on Coxeter groups, J. Algebra
**224**(2000), 151–165. - Fulman, J., Counting semisimple orbits of finite Lie algebras by genus,
*J. Algebra***217**(1999), 170-179. - Fulman, J., The combinatorics of biased riffle shuffles.
*Ann. of Combin.***2**(1998), 1-6. - 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.
- Fulman, J., Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting, to appear in J. Algebra.
- Phil Hanlon,
*The action of $S_n$ on the components of the Hodge decomposition of Hochschild homology*, Michigan Math. J.**37**(1990), no. 1, 105–124. MR**1042517**, DOI https://doi.org/10.1307/mmj/1029004069 - Humphreys, J.,
*Reflection groups and Coxeter groups*. Cambridge Studies in Advanced Mathematics**29**, Cambridge University Press, Cambridge. - Peter Orlik and Louis Solomon,
*Coxeter arrangements*, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 269–291. MR**713255** - Steven Shnider and Shlomo Sternberg,
*Quantum groups*, Graduate Texts in Mathematical Physics, II, International Press, Cambridge, MA, 1993. From coalgebras to Drinfel′d algebras; A guided tour. MR**1287162** - Shephard, G.C., and Todd, J.A., Finite unitary reflection groups
*Canadian J. Math.***6**(1954), 274-304. - Louis Solomon,
*The orders of the finite Chevalley groups*, J. Algebra**3**(1966), 376–393. MR**199275**, DOI https://doi.org/10.1016/0021-8693%2866%2990007-X - Stanley, R., Generalized riffle shuffles and quasi-symmetric functions, http://xxx.lanl.gov/abs/math.CO/9904042.
- Thomas Zaslavsky,
*Facing up to arrangements: face-count formulas for partitions of space by hyperplanes*, Mem. Amer. Math. Soc.**1**(1975), no. issue 1, 154, vii+102. MR**357135**, DOI https://doi.org/10.1090/memo/0154

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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

Article copyright:
© Copyright 2000
American Mathematical Society