Noncommutative symmetric functions and combinatorial Hopf algebras

  • Jean-Yves Thibon
Part of the Publications of the Scuola Normale Superiore book series (PSNS, volume 12)

Abstract

We present on a few examples a class of algebras which are increasingly popular in Combinatorics, and tend to permeate other fields as well. In particular, some of these algebras have connections with mould calculus and resurgence theory. They can be approached in many different ways. Here, they will be regarded as generalizations of the algebra of symmetric functions.

Keywords

Hopf Algebra Symmetric Function Primitive Element Free Associative Algebra Hopf Algebra Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Abe, “Hopf Algebras”, Cambridge Tracts in Mathematics, 74, Cambridge University Press, Cambridge-New York, 1980, xii+284.Google Scholar
  2. [2]
    M. Aguiar, N. Bergeron and F. Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math. 142 (2006), 1–30.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    A. Baker and B. Richter, Quasisymmetric functions from a topological point of view, Math. Scand. 103 (2008), 208–242.MathSciNetMATHGoogle Scholar
  4. [4]
    F. Bergeron, N. Bergeron and A. M. Garsia, Idempotents for the free Lie algebra and q-enumeration, In: “Invariant theory and tableaux”, D. Stanton (ed.), IMA Volumes in Mathematics and its Applications, Vol. 19, Springer, 1988.Google Scholar
  5. [5]
    N. Bergeron, F. Hivert and J.-Y. Thibon, The peak algebra and the Hecke-Clifford algebras at q = 0, J. Combin. Theor. A 117 (2004), 1–19.MathSciNetCrossRefGoogle Scholar
  6. [6]
    I. Bialynicki-Birula, B. Mielnik and J. Plebański, Explicit solution of the continuous Baker-Campbell-Hausdorff problem, Annals of Physics 51 (1969), 187–200.CrossRefGoogle Scholar
  7. [7]
    D. Blessenohl and H. Laue, Algebraic combinatorics related to the free Lie algebra, In: “Actes du 29-ième Séminaire Lotharingien de Combinatoire”, A. Kerber (ed), Publ. IRMA, Strasbourg, 1993, 1–21.Google Scholar
  8. [8]
    N. Bourbaki, Groupes et Algèbres de Lie, Chapter 2 et 3, Hermann, 1972.Google Scholar
  9. [9]
    C. Brouder and A. Frabetti, QED Hopf algebras on planar binary trees, J. Algebra 267 (2003), 298–322.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    C. Brouder, A. Frabetti and C. Krattenthaler, Noncommutative Hopf algebra of formal diffeomorphisms, Adv. Math. 200 (2006), 479–524.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    F. Chapoton, Algèbres de Hopf des permutahèdres, associahèdres et hypercubes, (French) [Hopf algebras of permutahedra, associahedra and hypercubes], Adv. Math. 150 (2000), 264–275.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    F. Chapoton, Construction de certaines opérades et bigèbres associ ées aux polytopes de Stasheff et hypercubes, Trans. Amer. Math. Soc. 354(1) (2002), 63–74 (electronic).MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    F. Chapoton, The anticyclic operad of moulds, Int. Math. Res. Not. IMRN 2007, no 20, Art. ID rnm078, 36 pp.Google Scholar
  14. [14]
    F. Chapoton, F. Hivert, J.-C. Novelli and J.-Y. Thibon, An operational calculus for the mould operad, Int. Math. Res. Not. IMRN 2008, no. 9, Art. ID rnn018, 22 pp.Google Scholar
  15. [15]
    A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), 203–242.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    G. Duchamp, F. Hivert and J.-Y. Thibon, Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, Internat. J. Alg. Comput. 12 (2002), 671–717.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    G. Duchamp, A. Klyachko, D. Krob and J.-Y. Thibon, Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras, Discrete Mathematics and Theoretical Computer Science 1 (1997), 159–216.MathSciNetMATHGoogle Scholar
  18. [18]
    G. Duchamp, D. Krob, B. Leclerc and J.-Y. Thibon, Déformations de projecteurs de Lie, C.R. Acad. Sci. Paris 319 (1994), 909–914.MathSciNetMATHGoogle Scholar
  19. [19]
    G. Duchamp, D. Krob and E. A. Vassilieva, Zassenhaus Lie idempotents, q-bracketing and a new exponential/logarithm correspondence, J. Algebraic Combin. 12 (2000), no. 3, 251–277.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    E.B. Dynkin, Calculation of the coefficients in the Campbell-Baker-Hausdorff formula, Dokl. Akad. Nauk. SSSR (N.S.) 57 (1947), 323–326 (in Russian).MathSciNetMATHGoogle Scholar
  21. [21]
    K. Ebrahimi-fard, D. Manchon and F. Patras A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov’s recursion, J. Noncommut. Geom. 3 (2009), 181–222.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    J. Ecalle, Les fonctions résurgentes, tome 1: Les algèbres de fonctions résurgentes, Publications mathématiques d’Orsay, 1981, volume 81-05.Google Scholar
  23. [23]
    J. Ecalle, Les fonctions résurgentes, tome 2: Les fonctions résurgentes appliquées à l’itération, Publications mathématiques d’Orsay, 1981, volume 81-06.Google Scholar
  24. [24]
    J. Ecalle, Les fonctions résurgentes, tome 3: L’équation du pont et la classifications analytique des objets locaux, Publications mathématiques d’Orsay, 1985, volume 85-05.Google Scholar
  25. [25]
    J. Ecalle and F. Menous, Well-behaved convolution averages and the non-accumulation theorem for limit-cycles, In: “The Stokes Phenomenon and Hilbert’s 16th Problem”, pages 71–102. World Scientific, 1996.Google Scholar
  26. [26]
    H. Figueroa and J. M. Gracia-Bondía, Combinatorial Hopf algebras in quantum field theory. I., Rev. Math. Phys. 17 (2005), no. 8, 881–976.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    L. Foissy, Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations, Adv. Math. 218 (2008), 136–162.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Symmetric Functions 2001: Surveys of Developments and Perspectives (S. Fomin Ed.), NATO Science Series II: Mathematics, Physics and Chemistry, vol. 74, Kluwer, Dordrecht, 2002.Google Scholar
  29. [29]
    K.O. Friedrichs, Mathematical aspects of the quantum theory of fields, Interscience, New-York, 1951.Google Scholar
  30. [30]
    A.M. Garsia, Combinatorics of the free Lie algebra and the symmetric group, In: “Analysis, et cetera...”, Jürgen Moser Festschrift, Academic press, New York, (1990), 309–82.Google Scholar
  31. [31]
    A.M. Garsia and C. Reutenauer, A decomposition of Solomon’s descent algebra Advances in Math. 77 (1989), 189–262.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    I. Gessel, Multipartite P-partitions and inner product of skew Schur functions, Contemp. Math. 34 (1984), 289–301.MathSciNetGoogle Scholar
  33. [33]
    P. Gordan, Symmetrische Functionen (German), Math. Ann. 52 (1899), 501–528.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    R. Grossman and R. G. Larson, Hopf-algebraic structure of families of trees, J. Algebra 126 (1989), 184–210.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    R. Grossman and R. G. Larson, Hopf-algebraic structure of combinatorial objects and differential operators, Israel J. Math. 72 (1990), 109–117.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh and J.-Y. Thibon, Noncommutative symmetric functions, Adv. in Math. 112 (1995), 218–348.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    I. Gessel, Multipartite P-partitions and inner product of skew Schur functions, Contemp. Math. 34 (1984), 289–301.MathSciNetGoogle Scholar
  38. [38]
    F. Hivert, J.-C. Novelli and J.-Y. Thibon, The algebra of binary search trees, Theoret. Comput. Sci. 339 (2005), 129–165.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    F. Hivert, J.-C. Novelli and J.-Y. Thibon, Yang-Baxter bases of 0-Hecke algebras and representation theory of 0-Ariki-Koike-Shoji algebras, Adv. Math. 205 (2006), 504–548.MathSciNetMATHCrossRefGoogle Scholar
  40. [40]
    F. Hivert, J.-C. Novelli and J.-Y. Thibon, Trees, functional equations, and combinatorial Hopf algebras, European J. Combin. 29 (2008), no. 7, 1682–1695.MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras, J. Alg. Comb. 28 (2008) n. 1, 65–95.MathSciNetMATHCrossRefGoogle Scholar
  42. [42]
    R. Holtkamp, On Hopf algebra structures over free operads, Adv. Math. 207 (2006), 544–565.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    A.A. Klyachko, Lie elements in the tensor algebra, Siberian Math. J. 15 (1974), 1296–1304.MATHGoogle Scholar
  44. [44]
    D.E. Knuth, “The art of computer programming”, vol.3: Sorting and searching, Addison-Wesley, 1973.Google Scholar
  45. [45]
    D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 (1998), 303–334.MathSciNetMATHGoogle Scholar
  46. [46]
    D. Krob, B. Leclerc and J.-Y. Thibon, Noncommutative symmetric functions II: Transformations of alphabets, Internat. J. Alg. Comput. 7 (1997), 181–264.MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    D. Krob and J.-Y. Thibon, Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at q = 0, J. Algebraic Combin. 6 (1997) n. 4, 339–376.MathSciNetMATHCrossRefGoogle Scholar
  48. [48]
    J.-L. Loday, Generalized bialgebras and triples of operads, Astrisque no. 320 (2008), x+116 pp.Google Scholar
  49. [49]
    J.-L. Loday and M. O. Ronco, Hopf algebra of the planar binary trees, Adv. Math., 139 (1998) n. 2, 293–309.MathSciNetMATHCrossRefGoogle Scholar
  50. [50]
    J.-L. Loday and M. O. Ronco, Trialgebras and families of polytopes, Contemporary Mathematics 346 (2004).Google Scholar
  51. [51]
    J.-L. Loday and M. O. Ronco, Combinatorial Hopf Algebras, to appear in the Proceedings of Alain Connes birthday Conference, Jan 2009. ArXiv: math:08100435.Google Scholar
  52. [52]
    I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, 1995.Google Scholar
  53. [53]
    W. Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math. VII (1954), 649–673.MathSciNetCrossRefGoogle Scholar
  54. [54]
    C. Malvenuto and C. Reutenauer, Duality between quasisymmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), 967–982.MathSciNetMATHCrossRefGoogle Scholar
  55. [55]
    F. Menous, Les bonnes moyennes uniformisantes et leurs applications à la resommation réelle, Thèse de doctorat, Orsay, 1996.Google Scholar
  56. [56]
    B. Mielnik and J. Plebański, Combinatorial approach to Baker-Campbell-Hausdorff exponents, Ann. Inst. Henri Poincaré, Section A, vol. XII (1970), 215–254.Google Scholar
  57. [57]
    J.-C. Novelli and J.-Y. Thibon, Hopf algebras and dendriform structures arising from parking functions, Fund. Math. 193 (2007), 189–241.MathSciNetMATHCrossRefGoogle Scholar
  58. [58]
    J.-C. Novelli and J.-Y. Thibon, Noncommutative symmetric functions and Lagrange inversion, Adv. in Appl. Math. 40 (2008), 8–35.MathSciNetMATHCrossRefGoogle Scholar
  59. [59]
    J.-C. Novelli and J.-Y. Thibon, Superization and (q, t)-specialization in combinatorial Hopf algebras, Electronic J. Combin. 16(2) (2009), R21, 46 pp.MathSciNetGoogle Scholar
  60. [60]
    F. Patras, L’algèbre des descentes d’une bigèbre graduée, J. Algebra 170 (1994), 547–566.MathSciNetMATHCrossRefGoogle Scholar
  61. [61]
    S. Poirier and C. Reutenauer, Algèbre de Hopf des tableaux, Ann. Sci. Math. Qébec 19 (1995), 79–90.MathSciNetMATHGoogle Scholar
  62. [62]
    R. Ree, Generalized Lie elements, Canad. J. Math. 12 (1960), 493–502.MathSciNetMATHCrossRefGoogle Scholar
  63. [63]
    C. Reutenauer, Theorem of Poincaré-Birkhoff-Witt, logarithm and representations of the symmetric group whose order are the Stirling numbers, In: “Combinatoire énumérative, Proceedings”, Montréal 1985 G. Labelle and P. Leroux (eds.), Lecture Notes in Math., 1234, Springer, (1986), 267–284.Google Scholar
  64. [64]
    C. Reutenauer, Free Lie algebras, Oxford, 1993.Google Scholar
  65. [65]
    L. Solomon, On the Poincaré-Birkhoff-Witt theorem, J. Comb. Theory 4 (1968), 363–375.MATHCrossRefGoogle Scholar
  66. [66]
    L. Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), 255–268.MathSciNetMATHCrossRefGoogle Scholar
  67. [67]
    N.J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (electronic), http://www.research.att.com/~njas/sequences/Google Scholar
  68. [68]
    W. Specht, Die linearen Beziehungen zwischen hÖheren Kommutatoren, Math. Zeit. 51 (1948), 367–376.MathSciNetMATHCrossRefGoogle Scholar
  69. [69]
    F. Wever, Über Invarianten in Lieschen Ringen, Math. Annalen 120 (1949), 563–580.MathSciNetMATHCrossRefGoogle Scholar
  70. [70]
    R.M. Wilcox, Exponential operators and parameter differentiation in quantum physics, J. Math. Phys. 8 (1967), 962–982.MathSciNetMATHCrossRefGoogle Scholar
  71. [71]
    A.V. Zelevinsky, Representations of finite classical groups. A Hopf algebra approach, Lecture Notes in Mathematics, 869. Springer-Verlag, Berlin-New York, 1981. iv+184 pp.Google Scholar

Copyright information

© Scuola Normale Superiore Pisa 2011

Authors and Affiliations

  • Jean-Yves Thibon
    • 1
  1. 1.Institut Gaspard MongeUniversité Paris-Est Marne-la-ValléeMarne-la-Vallée cedex 2France

Personalised recommendations