Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras
Samuel K. Hsiao
and Gizem Karaali
DOI: 10.1007/s10801-011-0279-3
Abstract
We develop a theory of multigraded (i.e., \Bbb N l -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.
Pages: 451–506
Keywords: keywords combinatorial Hopf algebra; multigraded Hopf algebra; quasisymmetric function; symmetric function; noncommutative symmetric function; Eulerian poset
Full Text: PDF
References
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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)
3. Aguiar, M., Sottile, F.: Structure of the Malvenuto-Reutenauer Hopf algebra of permutations. Adv. Math. 191(2), 225-275 (2005)
4. Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn- Sommerville relations. Compos. Math. 142(1), 1-30 (2006)
5. Aval, J.-C., Bergeron, F., Bergeron, N.: Diagonal Temperley-Lieb invariants and harmonics. Sémin. Lothar. Comb. 54A, B54Aq (2005/2007) 19 pp. (electronic)
6. Baumann, P., Hohlweg, C.: A Solomon descent theory for the wreath products G Sn. Trans. Am. Math. Soc. 360(3), 1475-1538 (2008) (electronic)
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)
8. Bergeron, N., Hohlweg, C.: Coloured peak algebras and Hopf algebras. J. Algebr. Comb. 24(3), 299- 330 (2006)
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 J Algebr Comb (2011) 34:451-506
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)
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)
13. Billera, L.J., Hsiao, S.K., van Willigenburg, S.: Peak quasisymmetric functions and Eulerian enumeration. Adv. Math. 176(2), 248-276 (2003)
14. Billera, L.J., Liu, N.: Noncommutative enumeration in graded posets. J. Algebr. Comb. 12(1), 7-24 (2000)
15. Billey, S., Haiman, M.: Schubert polynomials for the classical groups. J. Am. Math. Soc. 8(2), 443- 482 (1995)
16. Duchamp, G., Hivert, F., Thibon, J.-Y.: Noncommutative symmetric functions. VI. Free quasisymmetric functions and related algebras. Int. J. Algebra Comput. 12(5), 671-717 (2002)
17. Ehrenborg, R.: On posets and Hopf algebras. Adv. Math. 119(1), 1-25 (1996)
18. Ehrenborg, R.: k-Eulerian posets. Order 18(3), 227-236 (2001)
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)
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)
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)
23. Hsiao, S.K., Petersen, T.K.: Colored posets and colored quasisymmetric functions. Ann. Comb. (2010).
24. Krob, D., Leclerc, B., Thibon, J.-Y.: Noncommutative symmetric functions. II. Transformations of alphabets. Int. J. Algebra Comput. 7(2), 181-264 (1997)
25. Loday, J.-L.: On the algebra of quasi-shuffles. Manuscr. Math. 123(1), 79-93 (2007)
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)
27. Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(3), 967-982 (1995)
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)
29. Novelli, J.-C., Thibon, J.-Y.: Free quasi-symmetric functions of arbitrary level (2004). Available at
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
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)
32. Reutenauer, C.: Free Lie Algebras. London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press, New York (1993)
33. Schocker, M.: The peak algebra of the symmetric group revisited. Adv. Math. 192(2), 259-309 (2005)
34. Stanley, R.P.: Flag-symmetric and locally rank-symmetric partially ordered sets. Electron. J. Comb.
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