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Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras

Abstract

We develop a theory of multigraded (i.e., ℕl-graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar et al. (Compos. Math. 142:1–30, 2006). In particular we introduce the notion of canonical k-odd and k-even subalgebras associated with any multigraded combinatorial Hopf algebra, extending simultaneously the work of Aguiar et al. and Ehrenborg. Among our results are specific categorical results for higher level quasisymmetric functions, several basis change formulas, and a generalization of the descents-to-peaks map.

References

  1. Aguiar, M.: Infinitesimal Hopf algebras and the cd-index of polytopes. Discrete Comput. Geom. 27(1), 3–28 (2002). Geometric combinatorics (San Francisco, CA/Davis, CA, 2000)

    MathSciNet  Google Scholar 

  2. Aguiar, M., Hsiao, S.K.: Canonical characters on quasi-symmetric functions and bivariate Catalan numbers. J. Comb. 11(2), 15 (2004/2006) 34 pp. (electronic)

    MathSciNet  Google Scholar 

  3. Aguiar, M., Sottile, F.: Structure of the Malvenuto-Reutenauer Hopf algebra of permutations. Adv. Math. 191(2), 225–275 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  4. Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn–Sommerville relations. Compos. Math. 142(1), 1–30 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  5. Aval, J.-C., Bergeron, F., Bergeron, N.: Diagonal Temperley–Lieb invariants and harmonics. Sémin. Lothar. Comb. 54A, B54Aq (2005/2007) 19 pp. (electronic)

    MathSciNet  Google Scholar 

  6. Baumann, P., Hohlweg, C.: A Solomon descent theory for the wreath products \(G\wr \mathfrak{S}_{n}\). Trans. Am. Math. Soc. 360(3), 1475–1538 (2008) (electronic)

    MathSciNet  MATH  Article  Google Scholar 

  7. Bayer, M.M., Billera, L.J.: Generalized Dehn–Sommerville relations for polytopes, spheres and Eulerian partially ordered sets. Invent. Math. 79(1), 143–157 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  8. Bergeron, N., Hohlweg, C.: Coloured peak algebras and Hopf algebras. J. Algebr. Comb. 24(3), 299–330 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  9. Bergeron, N., Mykytiuk, S., Sottile, F., van Willigenburg, S.: Noncommutative Pieri operators on posets. J. Comb. Theory, Ser. A 91(1–2), 84–110 (2000). In memory of Gian-Carlo Rota

    MATH  Article  Google Scholar 

  10. Bergeron, N., Mykytiuk, S., Sottile, F., van Willigenburg, S.: Personal communication (2001)

  11. Bergeron, N., Mykytiuk, S., Sottile, F., van Willigenburg, S.: Shifted quasi-symmetric functions and the Hopf algebra of peak functions. Discrete Math. 246(1–3), 57–66 (2002). English, with English and French summaries, Formal power series and algebraic combinatorics (Barcelona, 1999)

    MathSciNet  MATH  Article  Google Scholar 

  12. Bergeron, N., Hivert, F., Thibon, J.-Y.: The peak algebra and the Hecke–Clifford algebras at q=0. J. Comb. Theory, Ser. A 107(1), 1–19 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  13. Billera, L.J., Hsiao, S.K., van Willigenburg, S.: Peak quasisymmetric functions and Eulerian enumeration. Adv. Math. 176(2), 248–276 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  14. Billera, L.J., Liu, N.: Noncommutative enumeration in graded posets. J. Algebr. Comb. 12(1), 7–24 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  15. Billey, S., Haiman, M.: Schubert polynomials for the classical groups. J. Am. Math. Soc. 8(2), 443–482 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  16. 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)

    MathSciNet  MATH  Article  Google Scholar 

  17. Ehrenborg, R.: On posets and Hopf algebras. Adv. Math. 119(1), 1–25 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  18. Ehrenborg, R.: k-Eulerian posets. Order 18(3), 227–236 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  19. 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)

    MathSciNet  MATH  Article  Google Scholar 

  20. Gessel, I.M.: Multipartite P-partitions and inner products of skew Schur functions. In: Combinatorics and Algebra, Boulder, CO, 1983. Contemp. Math., vol. 34, pp. 289–317. Amer. Math. Soc., Providence (1984)

    Google Scholar 

  21. Gessel, I.M.: Enumerative applications of symmetric functions. Sémin. Lothar. Comb. B17a (1987) 17pp.

  22. Hoffman, M.E.: Quasi-shuffle products. J. Algebr. Comb. 11(1), 49–68 (2000)

    MATH  Article  Google Scholar 

  23. Hsiao, S.K., Petersen, T.K.: Colored posets and colored quasisymmetric functions. Ann. Comb. (2010). doi:10.1007/s00026-010-0059-0

    MathSciNet  Google Scholar 

  24. Krob, D., Leclerc, B., Thibon, J.-Y.: Noncommutative symmetric functions. II. Transformations of alphabets. Int. J. Algebra Comput. 7(2), 181–264 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  25. Loday, J.-L.: On the algebra of quasi-shuffles. Manuscr. Math. 123(1), 79–93 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  26. MacMahon, P.A.: Combinatory Analysis. Dover Phoenix Editions, vols. I, II. Dover, Mineola (2004). (Bound in one volume) Reprint of An introduction to combinatory analysis (1920) and Combinatory analysis. Vols. I, II (1915, 1916)

    MATH  Google Scholar 

  27. Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(3), 967–982 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  28. Mantaci, R., Reutenauer, C.: A generalization of Solomon’s algebra for hyperoctahedral groups and other wreath products. Commun. Algebra 23(1), 27–56 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  29. Novelli, J.-C., Thibon, J.-Y.: Free quasi-symmetric functions of arbitrary level (2004). Available at http://arxiv.org/abs/math.CO/0405597

  30. Novelli, J.-C., Thibon, J.-Y.: Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions (2008). Available at http://arxiv.org/abs/0806.3682

  31. Poirier, S.: 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, pp. 315–343 (1998)

    Google Scholar 

  32. Reutenauer, C.: Free Lie Algebras. London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press, New York (1993)

    MATH  Google Scholar 

  33. Schocker, M.: The peak algebra of the symmetric group revisited. Adv. Math. 192(2), 259–309 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  34. Stanley, R.P.: Flag-symmetric and locally rank-symmetric partially ordered sets. Electron. J. Comb. 3(2), 6 (1996). Approx. 22 pp. (electronic). The Foata Festschrift

    MathSciNet  Google Scholar 

  35. Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (1997). With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original

    MATH  Google Scholar 

  36. Stembridge, J.R.: Enriched P-partitions. Trans. Am. Math. Soc. 349(2), 763–788 (1997)

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Samuel K. Hsiao.

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This work began while S.K. Hsiao was supported by an NSF Postdoctoral Research Fellowship.

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Hsiao, S.K., Karaali, G. Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras. J Algebr Comb 34, 451–506 (2011). https://doi.org/10.1007/s10801-011-0279-3

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  • DOI: https://doi.org/10.1007/s10801-011-0279-3

Keywords

  • Combinatorial Hopf algebra
  • Multigraded Hopf algebra
  • Quasisymmetric function
  • Symmetric function
  • Noncommutative symmetric function
  • Eulerian poset