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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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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 $A$,$B$,$C$,$H_3$, 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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  • [BaD] Dave Bayer and Persi Diaconis, Trailing the dovetail shuffle to its lair, Ann. Appl. Probab. 2 (1992), no. 2, 294–313. MR 1161056
  • [BB] 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,
  • [BBHT] 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,
  • [B3] 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,
  • [B] 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), 135-174. CMP 99:16
  • [BrD] Brown, K. and Diaconis, P., Random walk and hyperplane arrangements. Ann. of Probab. 26 (1998), 1813-1854. CMP 99:09
  • [C1] 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
  • [C2] R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59. MR 0318337
  • [F1] Fulman, J., Semisimple orbits of Lie algebras and card shuffling measures on Coxeter groups, J. Algebra 224 (2000), 151-165.
  • [F2] Fulman, J., Counting semisimple orbits of finite Lie algebras by genus, J. Algebra 217 (1999), 170-179. CMP 99:15
  • [F3] Fulman, J., The combinatorics of biased riffle shuffles. Ann. of Combin. 2 (1998), 1-6.
  • [F4] Fulman, J., Cellini's descent algebra, dynamical systems, and semisimple conjugacy classes of finite groups of Lie type,
  • [F5] Fulman, J., Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting, to appear in J. Algebra.
  • [Ha] 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,
  • [H] Humphreys, J., Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge.
  • [OS] 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
  • [SS] 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
  • [ST] Shephard, G.C., and Todd, J.A., Finite unitary reflection groups Canadian J. Math. 6 (1954), 274-304. MR 15:600b
  • [So1] Louis Solomon, The orders of the finite Chevalley groups, J. Algebra 3 (1966), 376–393. MR 0199275,
  • [St] Stanley, R., Generalized riffle shuffles and quasi-symmetric functions,
  • [Z] 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 0357135

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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