Journal of Theoretical Probability

, Volume 32, Issue 4, pp 1804–1844 | Cite as

Lumpings of Algebraic Markov Chains Arise from Subquotients

  • C. Y. Amy PangEmail author


A function on the state space of a Markov chain is a “lumping” if observing only the function values gives a Markov chain. We give very general conditions for lumpings of a large class of algebraically defined Markov chains, which include random walks on groups and other common constructions. We specialise these criteria to the case of descent operator chains from combinatorial Hopf algebras, and, as an example, construct a “top-to-random-with-standardisation” chain on permutations that lumps to a popular restriction-then-induction chain on partitions, using the fact that the algebra of symmetric functions is a subquotient of the Malvenuto–Reutenauer algebra.


Markov chain Random walks on groups Card shuffling Combinatorial Hopf algebras 

Mathematics Subject Classification (2010)

60J10 16T30 05E05 



I would like to thank Nathan Williams for a question that motivated this research, and Persi Diaconis, Jason Fulman and Franco Saliola for numerous helpful conversations, and Federico Ardila, Grégory Châtel, Mathieu Guay-Paquet, Simon Rubenstein-Salzedo, Yannic Vargas and Graham White for useful comments. SAGE computer software [65] was very useful, especially the combinatorial Hopf algebras coded by Aaron Lauve and Franco Saliola.


  1. 1.
    Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn–Sommerville relations. Compos. Math. 142(1), 1–30 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aguiar, M., Mahajan, S.: Monoidal functors, species and Hopf algebras, CRM Monograph Series, vol. 29. American Mathematical Society, Providence, RI (2010). With forewords by Kenneth Brown and Stephen Chase and André JoyalGoogle Scholar
  3. 3.
    Aguiar, M., Mahajan, S.: Hopf monoids in the category of species. In: Hopf Algebras and Tensor categories. Contemporary Mathematics, vol. 585, pp. 17–124 (2013).
  4. 4.
    Aguiar, M., Sottile, F.: Structure of the Malvenuto–Reutenauer Hopf algebra of permutations. Adv. Math. 191(2), 225–275 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aguiar, M., Sottile, F.: Structure of the Loday–Ronco Hopf algebra of trees. J. Algebra 295(2), 473–511 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Aldous, D., Diaconis, P.: Shuffling cards and stopping times. Am. Math. Mon. 93(5), 333–348 (1986). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Aldous, D., Diaconis, P.: Strong uniform times and finite random walks. Adv. Appl. Math. 8(1), 69–97 (1987). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Assaf, S., Diaconis, P., Soundararajan, K.: A rule of thumb for riffle shuffling. Ann. Appl. Probab. 21(3), 843–875 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Athanasiadis, C.A., Diaconis, P.: Functions of random walks on hyperplane arrangements. Adv. Appl. Math. 45(3), 410–437 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ayyer, A., Schilling, A., Steinberg, B., Thiéry, N.M.: Markov chains, \(\cal{R}\)-trivial monoids and representation theory. Int. J. Algebra Comput. 25(1–2), 169–231 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bayer, D., Diaconis, P.: Trailing the dovetail shuffle to its lair. Ann. Appl. Probab. 2(2), 294–313 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bergeron, N., Li, H.: Algebraic structures on Grothendieck groups of a tower of algebras. J. Algebra 321(8), 2068–2084 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bernstein, M.: A random walk on the symmetric group generated by random involutions. ArXiv e-prints (2016)Google Scholar
  14. 14.
    Bidigare, P., Hanlon, P., Rockmore, D.: A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements. Duke Math. J. 99(1), 135–174 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Britnell, J.R., Wildon, M.: Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin types A, B and D. J. Combin. Theory Ser. A 148, 116–144 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Brown, K.S.: Semigroups, rings, and Markov chains. J. Theor. Probab. 13(3), 871–938 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Corteel, S., Williams, L.K.: A Markov chain on permutations which projects to the PASEP. Int. Math. Res. Not. IMRN (17), Art. ID rnm055, 27 (2007).
  18. 18.
    Diaconis, P.: Group representations in probability and statistics. In: Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 11. Institute of Mathematical Statistics, Hayward, CA (1988)Google Scholar
  19. 19.
    Diaconis, P., Fill, J.A., Pitman, J.: Analysis of top to random shuffles. Combin. Probab. Comput. 1(2), 135–155 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Diaconis, P., Holmes, S.P.: Random walks on trees and matchings. Electron. J. Probab. 7(6), 17 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Diaconis, P., Mayer-Wolf, E., Zeitouni, O., Zerner, M.P.W.: The Poisson–Dirichlet law is the unique invariant distribution for uniform split-merge transformations. Ann. Probab. 32(1B), 915–938 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Diaconis, P., Pang, C.Y.A., Ram, A.: Hopf algebras and Markov chains: two examples and a theory. J. Algebraic Combin. 39(3), 527–585 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Diaconis, P., Ram, A.: Analysis of systematic scan Metropolis algorithms using Iwahori–Hecke algebra techniques. Michigan Math. J. 48, 157–190 (2000). Dedicated to William Fulton on the occasion of his 60th birthdayMathSciNetCrossRefGoogle Scholar
  24. 24.
    Diaconis, P., Shahshahani, M.: Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57(2), 159–179 (1981). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Diaconis, P., Shahshahani, M.: Time to reach stationarity in the Bernoulli–Laplace diffusion model. SIAM J. Math. Anal. 18(1), 208–218 (1987). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Doob, J.L.: Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85, 431–458 (1957)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Duchamp, G., Hivert, F., Thibon, J.Y.: Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras. Int. J. Algebra Comput. 12(5), 671–717 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Duchi, E., Schaeffer, G.: A combinatorial approach to jumping particles. J. Combin. Theory Ser. A 110(1), 1–29 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Durrett, R., Granovsky, B.L., Gueron, S.: The equilibrium behavior of reversible coagulation–fragmentation processes. J. Theor. Probab. 12(2), 447–474 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Fill, J.A.: An exact formula for the move-to-front rule for self-organizing lists. J. Theor. Probab. 9(1), 113–160 (1996). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Fulman, J.: Card shuffling and the decomposition of tensor products. Pac. J. Math. 217(2), 247–262 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Fulman, J.: Commutation relations and Markov chains. Probab. Theory Relat. Fields 144(1–2), 99–136 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Fulton, W.: Young tableaux. In: London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997). With applications to representation theory and geometryGoogle Scholar
  34. 34.
    Gelfand, I.M., Krob, D., Lascoux, A., Leclerc, B., Retakh, V.S., Thibon, J.Y.: Noncommutative symmetric functions. Adv. Math. 112(2), 218–348 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gessel, I.M.: Multipartite \(P\)-partitions and inner products of skew Schur functions. In: Combinatorics and algebra (Boulder, Colo., 1983), Contemp. Math., vol. 34, pp. 289–317. Amer. Math. Soc., Providence, RI (1984).
  36. 36.
    Giraudo, S.: Algebraic and combinatorial structures on pairs of twin binary trees. J. Algebra 360, 115–157 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Greene, C., Nijenhuis, A., Wilf, H.S.: A probabilistic proof of a formula for the number of Young tableaux of a given shape. Adv. Math. 31(1), 104–109 (1979). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Greene, C., Nijenhuis, A., Wilf, H.S.: Another probabilistic method in the theory of Young tableaux. J. Combin. Theory Ser. A 37(2), 127–135 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Grinberg, D., Reiner, V.: Hopf algebras in combinatorics. ArXiv e-prints (2014)Google Scholar
  40. 40.
    Hivert, F.: An introduction to combinatorial Hopf algebras—examples and realizations. In: Physics and theoretical computer science, NATO Secur. Sci. Ser. D Inf. Commun. Secur., vol. 7, pp. 253–274. IOS, Amsterdam (2007)Google Scholar
  41. 41.
    Hivert, F., Luque, J.G., Novelli, J.C., Thibon, J.Y.: The \((1-{\mathbb{E}})\)-transform in combinatorial Hopf algebras. J. Algebraic Combin. 33(2), 277–312 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Joni, S.A., Rota, G.C.: Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61(2), 93–139 (1979)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Kemeny, J.G., Snell, J.L.: Finite Markov Chains. The University Series in Undergraduate Mathematics. D. Van Nostrand Co., Inc., Princeton (1960)zbMATHGoogle Scholar
  44. 44.
    Kemeny, J.G., Snell, J.L., Knapp, A.W.: Denumerable Markov Chains. D. Van Nostrand Co., Inc., Princeton (1966)zbMATHGoogle Scholar
  45. 45.
    Krob, D., Thibon, J.Y.: Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at \(q=0\). J. Algebraic Combin. 6(4), 339–376 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Letac, G., Takács, L.: Random walks on an \(m\)-dimensional cube. J. Reine Angew. Math. 310, 187–195 (1979)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov chains and mixing times. American Mathematical Society, Providence, RI With a Chapter by James G. Propp and David B, Wilson (2009)Google Scholar
  48. 48.
    Loday, J.L., Ronco, M.O.: Hopf algebra of the planar binary trees. Adv. Math. 139(2), 293–309 (1998). MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Lothaire, M.: Combinatorics on Words. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1997). With a foreword by Roger Lyndon and a preface by Dominique Perrin; corrected reprint of the 1983 original, with a new preface by PerrinGoogle Scholar
  50. 50.
    Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(3), 967–982 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Pang, C.Y.A.: A Hopf-power Markov chain on compositions. In: 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), Discrete Math. Theor. Comput. Sci. Proc., AS, pp. 499–510. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2013)Google Scholar
  52. 52.
    Pang, C.Y.A.: Hopf algebras and Markov chains. ArXiv e-prints (2014). A revised thesisGoogle Scholar
  53. 53.
    Pang, C.Y.A.: Card-shuffling via convolutions of projections on combinatorial Hopf algebras. In: 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), Discrete Math. Theor. Comput. Sci. Proc., AU, pp. 49–60. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2015)Google Scholar
  54. 54.
    Pang, C.Y.A.: Markov chains from descent operators on combinatorial Hopf algebras. ArXiv e-prints (2016). References are to a second version in preparationGoogle Scholar
  55. 55.
    Patras, F.: L’algèbre des descentes d’une bigèbre graduée. J. Algebra 170(2), 547–566 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Phatarfod, R.M.: On the matrix occurring in a linear search problem. J. Appl. Probab. 28(2), 336–346 (1991)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Poirier, S., Reutenauer, C.: Algèbres de Hopf de tableaux. Ann. Sci. Math. Québec 19(1), 79–90 (1995)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Priez, J.B.: A lattice of combinatorial Hopf algebras, application to binary trees with multiplicities. ArXiv e-prints (2013)Google Scholar
  59. 59.
    Reutenauer, C.: Free Lie Algebras, London Mathematical Society Monographs. New Series, vol. 7. The Clarendon Press Oxford University Press, New York (1993). Oxford Science PublicationsGoogle Scholar
  60. 60.
    Rogers, L.C.G., Pitman, J.W.: Markov functions. Ann. Probab. 9(4), 573–582 (1981)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Sagan, B.E.: The symmetric group, Graduate Texts in Mathematics, vol. 203, second edn. Springer-Verlag, New York (2001). Representations, combinatorial algorithms, and symmetric functionsCrossRefGoogle Scholar
  62. 62.
    Saloff-Coste, L.: Random walks on finite groups. In: Kesten, H. (ed.) Probability on discrete structures, Encyclopaedia Math. Sci., vol. 110, pp. 263–346. Springer, Berlin (2004). CrossRefGoogle Scholar
  63. 63.
    Solomon, L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41(2), 255–264 (1976)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Stanley, R.P.: Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999). With a foreword by Gian-Carlo Rota and appendix 1 by Sergey FominGoogle Scholar
  65. 65.
    Stein, W., et al.: Sage Mathematics Software (Version 6.6). The Sage Development Team (2015).
  66. 66.
    Swart, J.: Advanced topics in Markov chains (2012). Lecture notes from a course at Charles University
  67. 67.
    Tsetlin, M.L.: Finite automata and models of simple forms of behaviour. Russ. Math. Surv. 18(4), 1–28 (1963)CrossRefGoogle Scholar
  68. 68.
    Zhao, Y.: Biased riffle shuffles, quasisymmetric functions, and the RSK algorithm (2009).
  69. 69.
    Zhou, H.: Examples of multivariate Markov chains with orthogonal polynomial eigenfunctions. ProQuest LLC, Ann Arbor, MI (2008). Thesis (Ph.D.)–Stanford UniversityGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Combinatoire et d’Informatique MathématiqueUniversité du Québec à MontréalMontrealCanada
  2. 2.Department of MathematicsHong Kong Baptist UniversityKowloonHong Kong

Personalised recommendations