Abstract
We construct bases of quasi-symmetric functions whose product rule is given by the shuffle of binary words, as for multiple zeta values in their integral representations, and then extend the construction to the algebra of free quasi-symmetric functions colored by positive integers. As a consequence, we show that the fractions introduced in Guo and Xie (Ramanujan J 25:307–317, 2011) provide a realization of this algebra by rational moulds extending that of free quasi-symmetric functions given in Chapoton et al. (Int Math Res Not IMRN 2008, no. 9, Art. ID rnn018, 2008).
Similar content being viewed by others
References
Bergeron, N., Reutenauer, C., Rosas, M., Zabrocki, M.: Invariants and coinvariants of the symmetric group in noncommuting variables, preprint arXiv:math.CO/0502082
Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisonĕk, P.: Special values of multiple polylogarithms. Trans. Am. Math. Soc. 353, 907–941 (2001)
Chapoton, F., Hivert, F., Novelli, J.-C., Thibon, J.-Y.: An operational calculus for the Mould operad. Int. Math. Res. Not. IMRN 2008, no. 9, Art. ID rnn018 (2008)
Duchamp, G., Hivert, F., Thibon, J.-Y.: Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras. Int. J. Algebra Comput. 12, 671–717 (2002)
Ecalle, J.: The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles. In: Costin, O., Fauvet, F., Menous, F., Sauzin, D. (eds.) Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation. Ann. Scuo. Norm. Pisa, vol. 2. Edizioni della Normale, Pisa (2011)
Gelfand, I.M., Krob, D., Lascoux, A., Leclerc, B., Retakh, V.S., Thibon, J.-Y.: Noncommutative symmetric functions. Adv. Math. 112, 218–348 (1995)
Gessel, I.: Multipartite \(P\)-partitions and inner product of skew Schur functions. Contemp. Math. 34, 289–301 (1984)
Guo, L., Keigher, B.: Baxter algebras and shuffle product. Adv. Math. 151, 101–127 (2000)
Guo, L., Xie, B.: The shuffle relation of fractions from multiple zeta values. Ramanujan J. 25, 307–317 (2011)
Hivert, F.: Hecke algebras, difference operators, and quasi-symmetric functions. Adv. Math. 155, 181–238 (2000)
Hivert, F., Novelli, J.-C., Thibon, J.-Y.: Commutative combinatorial Hopf algebras. J. Algebraic Combin. 28(1), 65–95 (2008)
Hoffman, M.E.: Quasi-shuffle products. J. Algebraic Comb. 11, 49–68 (2000)
Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177, 967–982 (1995)
Novelli, J.-C., Thibon, J.-Y.: Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions. Discret. Math. 310, 3584–3606 (2010)
Racinet, G.: Séries génératrices non-commutatives de polyzêtas et associateurs de Drinfeld. Doctoral Thesis, Université de Picardie Jules Verne, 2000. Available at http://tel.archives-ouvertes.fr/tel-00110891
Radford, D.E.: A natural ring basis for the shuffle algebra and an application to group schemes. J. Algebra 58, 432–454 (1979)
Schimming, R., Rida, S.Z.: Noncommutative Bell polynomials. Int. J. Algebra Comput. 6, 635–644 (1996)
Thibon, J.-Y., Ung, B.-C.-V.: Quantum quasi-symmetric functions and Hecke algebras. J. Phys. A: Math. Gen. 29, 7337–7348 (1996)
Zagier, D.: Values of Zeta Functions and Their Applications. First European Congress of Mathematics, Vol. II (Paris, 1992), Prog. Math., vol. 120, pp. 497–512. Birkhäuser, Basel (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research has been partially supported by the Project CARMA (ANR-12-BS01-0017) of the Agence Nationale de la Recherche and the Labex Bézout of Université Paris-Est.
Rights and permissions
About this article
Cite this article
Novelli, JC., Thibon, JY. Binary shuffle bases for quasi-symmetric functions. Ramanujan J 40, 207–225 (2016). https://doi.org/10.1007/s11139-016-9777-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-016-9777-1