Descent algebras, hyperplane arrangements, and shuffling cards
HTML articles powered by AMS MathViewer
- by Jason Fulman PDF
- Proc. Amer. Math. Soc. 129 (2001), 965-973 Request permission
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.References
- 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 10.1016/0022-4049(92)90153-7
- 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 10.1023/A:1022481230120
- 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 10.1016/0747-7171(92)90032-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 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
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- Louis Solomon, The orders of the finite Chevalley groups, J. Algebra 3 (1966), 376–393. MR 199275, DOI 10.1016/0021-8693(66)90007-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 10.1090/memo/0154
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
- © 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