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Journal of Theoretical Probability

, Volume 13, Issue 3, pp 871–938 | Cite as

Semigroups, Rings, and Markov Chains

  • Kenneth S. Brown
Article

Abstract

We analyze random walks on a class of semigroups called “left-regular bands.” These walks include the hyperplane chamber walks of Bidigare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are “generalized derangement numbers,” which may be of independent interest.

random walk Markov chain semigroup hyperplane arrangement diagonalization matroid derangement number 

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REFERENCES

  1. 1.
    Abels, H. (1991). The geometry of the chamber system of a semimodular lattice. Order 8(2), 143–158.Google Scholar
  2. 2.
    Aldous, D. (1999). The Moran process as a Markov chain on leaf-labeled trees, preprint.Google Scholar
  3. 3.
    Bayer, D., and Diaconis, P. (1992). Trailing the dovetail shuffle to its lair. Ann. Appl. Probab. 2(2), 294–313.Google Scholar
  4. 4.
    Bergeron, F., Bergeron, N., Howlett, R. B., and Taylor, D. E. (1992). A decomposition of the descent algebra of a finite Coxeter group. J. Algebraic Combin. 1(1), 23–44.Google Scholar
  5. 5.
    Bidigare, T. P. (1997). Hyperplane Arrangement Face Algebras and Their Associated Markov Chains, Ph.D. thesis, University of Michigan.Google Scholar
  6. 6.
    Bidigare, T. P., Hanlon, P., and Rockmore, D. N. (1999). A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements. Duke Math. J. 99(1), 135–174.Google Scholar
  7. 7.
    Billera, L. J., Brown, K. S., and Diaconis, P. (1999). Random walks and plane arrangements in three dimensions. Amer. Math. Monthly 106(6), 502–524.Google Scholar
  8. 8.
    Billera, L. J., and Liu, N. Noncommutative enumeration in graded posets. J. Algebraic Combin., to appear.Google Scholar
  9. 9.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., and Ziegler, G. M. (1993). Oriented Matroids, Encyclopedia of Mathematics and Its Applications, Vol. 46, Cambridge University Press, Cambridge.Google Scholar
  10. 10.
    Brown, K. S. (1989). Buildings, Springer-Verlag, New York.Google Scholar
  11. 11.
    Brown, K. S., and Diaconis, P. (1998). Random walks and hyperplane arrangements. Ann. Probab. 26(4), 1813–1854.Google Scholar
  12. 12.
    Désarménien, J. (1983). Une autre interprétation du nombre de dérangements. Sém. Lothar. Combin. 8, Art. B08b, 6 pp. (electronic).Google Scholar
  13. 13.
    Désarménien, J., and Wachs, M. L. (1993). Descent classes of permutations with a given number of fixed points. J. Combin. Theory Ser. A 64(2), 311–328.Google Scholar
  14. 14.
    Diaconis, P. From shuffling cards to walking around the building: An introduction to modern Markov chain theory, Proceedings of the 1998 International Congress of Mathematicians, to appear.Google Scholar
  15. 15.
    Diaconis, P. (1988). Group Representations in Probability and Statistics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, Vol. 11, Institute of Mathematical Statistics, Hayward, CA.Google Scholar
  16. 16.
    Diaconis, P., Fill, J. A., and Pitman, J. (1992). Analysis of top to random shuffles. Combin. Probab. Comput. 1(2), 135–155.Google Scholar
  17. 17.
    Diaconis, P., and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41(1), 45–76 (electronic).Google Scholar
  18. 18.
    Fill, J. A. (1996). An exact formula for the move-to-front rule for self-organizing lists. J. Theoret. Probab. 9(1), 113–160.Google Scholar
  19. 19.
    Greene, C. (1973). On the Möbius algebra of a partially ordered set. Advances in Math. 10, 177–187.Google Scholar
  20. 20.
    Grillet, P. A. (1995). Semigroups. An Introduction to the Structure Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 193, Marcel Dekker Inc., New York.Google Scholar
  21. 21.
    Grove, L. C., and Benson, C. T. (1985). Finite Reflection Groups, second ed., Graduate Texts in Mathematics, Vol. 99, Springer-Verlag, New York.Google Scholar
  22. 22.
    Högnäs, G., and Mukherjea, A. (1995). Probability Measures on Semigroups, Plenum Press, New York, Convolution products, random walks, and random matrices.Google Scholar
  23. 23.
    Humphreys, J. E. (1990). Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge.Google Scholar
  24. 24.
    Klein-Barmen, F. (1940). Über eine weitere Verallgemeinerung des Verbandsbegriffes. Math. Z. 46, 472–480.Google Scholar
  25. 25.
    Orlik, P., and Terao, H. (1992). Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften, Vol. 300, Springer-Verlag, Berlin.Google Scholar
  26. 26.
    Petrich, M. (1971). A construction and a classification of bands. Math. Nachr. 48, 263–274.Google Scholar
  27. 27.
    Petrich, M. (1977). Lectures in Semigroups, Wiley, London/New York/Sydney.Google Scholar
  28. 28.
    Phatarfod, R. M. (1991). On the matrix occurring in a linear search problem. J. Appl. Probab. 28(2), 336–346.Google Scholar
  29. 29.
    Schützenberger, M.-P. (1947). Sur certains treillis gauches. C. R. Acad. Sci. Paris 224, 776–778.Google Scholar
  30. 30.
    Serre, J.-P. (1977). Linear Representations of Finite Groups, Springer-Verlag, New York; Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42.Google Scholar
  31. 31.
    Solomon, L. (1967). The Burnside algebra of a finite group. J. Combin. Theory 2, 603–615.Google Scholar
  32. 32.
    Solomon, L. (1976). A Mackey formula in the group ring of a Coxeter group. J. Algebra 41(2), 255–264.Google Scholar
  33. 33.
    Stanley, R. P. (1996). Combinatorics and Commutative Algebra, second ed., Progress in Mathematics, Vol. 41, Birkhäuser Boston Inc., Boston, MA.Google Scholar
  34. 34.
    Stanley, R. P. (1997). Enumerative Combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, Vol. 49, Cambridge University Press, Cambridge, with a foreword by Gian-Carlo Rota, corrected reprint of the 1986 original.Google Scholar
  35. 35.
    Tits, J. (1974). Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin.Google Scholar
  36. 36.
    Wachs, M. L. (1989). On q-derangement numbers. Proc. Amer. Math. Soc. 106(1), 273–278.Google Scholar
  37. 37.
    Welsh, D. J. A. (1976). Matroid Theory, L. M. S. Monographs, No. 8, Academic Press [Harcourt Brace Jovanovich Publishers], London.Google Scholar
  38. 38.
    Whitney, H. (1935). On the Abstract Properties of Linear Dependence, Amer. J. Math., pp. 509–533; Collected Papers, Vol. I, 147–171.Google Scholar
  39. 39.
    Zaslavsky, T. (1975). Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes. Mem. Amer. Math. Soc. 1(154), vii+102 pp.Google Scholar
  40. 40.
    Zaslavsky, T. (1977). A combinatorial analysis of topological dissections. Advances in Math. 25(3), 267–285.Google Scholar
  41. 41.
    Ziegler, G. M. (1995). Lectures on Polytopes, Graduate Texts in Mathematics, Vol. 152, Springer-Verlag, New York.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Kenneth S. Brown
    • 1
  1. 1.Department of MathematicsCornell UniversityIthaca

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