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A Solomon descent theory for the wreath products $ G\wr\mathfrak{S}_n$


Authors: Pierre Baumann and Christophe Hohlweg
Journal: Trans. Amer. Math. Soc. 360 (2008), 1475-1538
MSC (2000): Primary 16S99; Secondary 05E05, 05E10, 16S34, 16W30, 20B30, 20E22
DOI: https://doi.org/10.1090/S0002-9947-07-04237-7
Published electronically: October 22, 2007
MathSciNet review: 2357703
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Abstract | References | Similar Articles | Additional Information

Abstract: We propose an analogue of Solomon's descent theory for the case of a wreath product $ G\wr\mathfrak{S}_n$, where $ G$ is a finite abelian group. Our construction mixes a number of ingredients: Mantaci-Reutenauer algebras, Specht's theory for the representations of wreath products, Okada's extension to wreath products of the Robinson-Schensted correspondence, and Poirier's quasisymmetric functions. We insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group.


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  • 1. R. M. Adin, F. Brenti and Y. Roichman, Descent representations and multivariate statistics, Trans. Amer. Math. Soc. 357 (2005), 3051-3082. MR 2135735 (2006e:20019)
  • 2. M. Aguiar, N. Bergeron and K. Nyman, The peak algebra and the descent algebras of type B and D, Trans. Amer. Math. Soc. 356 (2004), 2781-2824. MR 2052597 (2005e:20016)
  • 3. M. Aguiar, N. Bergeron and F. Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math. 142 (2006), 1-30. MR 2196760 (2006h:05233)
  • 4. M. D. Atkinson, A new proof of a theorem of Solomon, Bull. London Math. Soc. 18 (1986), 351-354. MR 838800 (87f:20043)
  • 5. J. C. Aval, F. Bergeron and N. Bergeron, Diagonal Temperley-Lieb invariants and harmonics, Sém. Lothar. Combin. 54A (2005/07), Art. B54Aq (electronic). MR 2264937
  • 6. E. Bagno and R. Biagioli, Colored-descent representations of complex reflection groups $ G(r,p,n)$, preprint arXiv:math.CO/0503238.
  • 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), 23-44. MR 1162640 (93g:20079)
  • 8. D. Blessenohl, C. Hohlweg and M. Schocker, A symmetry of the descent algebra of a finite Coxeter group, Adv. Math. 193 (2005), 416-437. MR 2137290 (2005m:20089)
  • 9. D. Blessenohl and M. Schocker, Noncommutative character theory of the symmetric group. River Edge: World Scientific, 2005.
  • 10. C. Bonnafé and C. Hohlweg, Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups, Ann. Inst. Fourier (Grenoble) 56 (2006), 131-181. MR 2228684 (2007e:20015)
  • 11. C. W. Curtis and I. Reiner, Methods of representation theory, Vol. I, Pure and Applied Mathematics. New York: John Wiley & Sons Inc., 1981. MR 0632548 (82i:20001)
  • 12. G. Duchamp, F. Hivert and J.-Y. Thibon, Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras, Internat. J. Algebra Comput. 12 (2002), 671-717. MR 1935570 (2003j:05126)
  • 13. I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh and J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995), 218-348. MR 1327096 (96e:05175)
  • 14. I. M. Gessel, Multipartite $ P$-partitions and inner products of skew Schur functions, in Combinatorics and algebra (Boulder, 1984), pp. 289-301, Contemp. Math., vol. 34, Providence: American Mathematical Society, 1984. MR 777705 (86k:05007)
  • 15. 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. MR 2258265
  • 16. A. Jöllenbeck, Nichtkommutative Charaktertheorie der symmetrischen Gruppen, Bayreuther Math. Schr. 56 (1999), 1-41. MR 1717091 (2000g:20022)
  • 17. A. Jöllenbeck and C. Reutenauer, Eine Symmetrieeigenschaft von Solomons Algebra und der höheren Lie-Charaktere, Abh. Math. Sem. Univ. Hamburg 71 (2001), 105-111. MR 1872717 (2002m:20021)
  • 18. R. Kilmoyer, Some irreducible complex representations of a finte group with a $ BN$-pair, Ph.D. dissertation, M.I.T., Cambridge, 1969.
  • 19. D. E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709-727. MR 0272654 (42:7535)
  • 20. J.-L. Loday and M. O. Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139 (1998), 293-309. MR 1654173 (99m:16063)
  • 21. I. G. Macdonald, Polynomial functors and wreath products, J. Pure Appl. Algebra 18 (1980), 173-204. MR 585222 (83j:15023)
  • 22. -, Symmetric functions and Hall polynomials, second ed., Oxford mathematical monographs, Oxford: Oxford University Press, 1995. MR 1354144 (96h:05207)
  • 23. C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), 967-982. MR 1358493 (97d:05277)
  • 24. R. Mantaci and C. Reutenauer, A generalization of Solomon's algebra for hyperoctahedral groups and other wreath products, Comm. Algebra 23 (1995), 27-56. MR 1311773 (95k:05182)
  • 25. S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Providence: American Mathematical Society, 1993. MR 1243637 (94i:16019)
  • 26. W. D.  Nichols, Bialgebras of type one, Comm. Algebra 6 (1978), 1521-1552. MR 0506406 (58:22150)
  • 27. J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions of arbitrary level, preprint arXiv:math.CO/0405597.
  • 28. S. Okada, Wreath products by the symmetric groups and product posets of Young's lattices, J. Combin. Theory Ser. A 55 (1990), 14-32. MR 1070012 (91i:20014)
  • 29. S. Poirier, Cycle type and descent set in wreath products, in Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995), Discrete Math. 180 (1998), 315-343. MR 1603753 (99i:20014)
  • 30. S. Poirier and C. Reutenauer, Algèbres de Hopf de tableaux, Ann. Sci. Math. Québec 19 (1995), 79-90. MR 1334836 (96g:05146)
  • 31. C. Reutenauer, Free Lie algebras, London Mathematical Society monographs new series, Oxford: Oxford University Press, 1993. MR 1231799 (94j:17002)
  • 32. M. Rosso, Quantum groups and quantum shuffles, Invent. Math. 133 (1998), 399-416. MR 1632802 (2000a:17021)
  • 33. L. Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), 255-264. MR 0444756 (56:3104)
  • 34. J. R. Stembridge, Enriched $ P$-partitions, Trans. Amer. Math. Soc. 349 (1997), 763-788. MR 1389788 (97f:06006)
  • 35. J.-Y. Thibon, Lectures on noncommutative symmetric functions, in Interaction of combinatorics and representation theory, pp. 39-94, Math. Soc. Japan Memoirs, vol. 11, Tokyo: The Mathematical Society of Japan, 2001. MR 1862149 (2003e:05142)
  • 36. J. Tits, Buildings of spherical type and finite BN-pairs. Lecture Notes in Mathematics, vol. 386, Berlin and New York: Springer-Verlag, 1974. MR 0470099 (57:9866)
  • 37. A. V. Zelevinsky, Representations of finite classical groups. A Hopf algebra approach. Lecture Notes in Mathematics, vol. 869, Berlin and New York: Springer-Verlag, 1981. MR 643482 (83k:20017)

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Additional Information

Pierre Baumann
Affiliation: Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
Email: baumann@math.u-strasbg.fr

Christophe Hohlweg
Affiliation: The Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1
Address at time of publication: Université du Québec à Montréal, Case postale 8888, succursale Centre-ville, Montréal, Québec, Canada H3C 3P8
Email: chohlweg@fields.utoronto.ca

DOI: https://doi.org/10.1090/S0002-9947-07-04237-7
Keywords: Wreath products, Solomon descent algebra, quasisymmetric functions.
Received by editor(s): April 1, 2005
Received by editor(s) in revised form: December 2, 2005
Published electronically: October 22, 2007
Additional Notes: This work was partially supported by Canada Research Chairs
Article copyright: © Copyright 2007 American Mathematical Society

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