×

The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer. (English) Zbl 1448.05209

Summary: This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra \(\mathfrak{S}\text{Sym}\) of permutations, mapped onto QSym by taking descents of permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions.
We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra \(\mathfrak{H}\text{Sym}\), linearly spanned by signed permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a permutation from a signed permutation defines a Hopf algebra surjection form \(\mathfrak{H}\text{Sym}\) to \(\mathfrak{S}\text{Sym}\) and taking a suitable descent from a signed permutation defines a linear surjection from \(\mathfrak{H}\text{Sym}\) to RQSym. The notion of weak \(P\)-partitions from signed permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from \(\mathfrak{S}\text{Sym}\) and RQSym to QSym, we obtain a commutative diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, permutations and signed permutations.

MSC:

05E05 Symmetric functions and generalizations
05A05 Permutations, words, matrices
16T30 Connections of Hopf algebras with combinatorics
16T05 Hopf algebras and their applications
20B30 Symmetric groups
05A17 Combinatorial aspects of partitions of integers
06A07 Combinatorics of partially ordered sets
16W99 Associative rings and algebras with additional structure
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aguiar, M.; Bergeron, N.; Nyman, K., The peak algebra and the descent algebras of types B and D, Trans. Am. Math. Soc., 356, 2781-2824 (2004) · Zbl 1043.05115
[2] Aguiar, M.; Bergeron, N.; Sottile, F., Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math., 142, 1-30 (2006) · Zbl 1092.05070
[3] Aguiar, M.; Sottile, F., Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Adv. Math., 191, 225-275 (2005) · Zbl 1056.05139
[4] Baker, A.; Richter, B., Quasisymmetric functions from a topological point of view, Math. Scand., 103, 208-242 (2008) · Zbl 1169.05049
[5] Bergeron, F.; Bergeron, N., Orthogonal idempotents in the descent algebra of \(B_n\) and applications, J. Pure Appl. Algebra, 79, 109-129 (1992) · Zbl 0793.20004
[6] Billera, L.; Hsiao, S.; van Willigenburg, S., Peak quasisymmetric functions and Eulerian enumeration, Adv. Math., 176, 248-276 (2003) · Zbl 1027.05105
[7] Borwein, J.; Bradley, D.; Broadhurst, D.; Lisoněk, P., Special values of multiple polylogarithms, Trans. Am. Math. Soc., 353, 907-941 (2001) · Zbl 1002.11093
[8] Björner, A.; Brenti, F., Combinatorics of Coxeter Groups (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1110.05001
[9] Cartier, P., On the structure of free Baxter algebras, Adv. Math., 9, 253-265 (1972) · Zbl 0267.60052
[10] Chow, C.-O., Noncommutative symmetric functions of type B (2001), MIT, Ph.D. thesis
[11] Clavier, P.; Guo, L.; Paycha, S.; Zhang, B., Renormalisation and locality: branched zeta values, IRMA Lect. Math. Theor. Phys., 32, 85-132 (2020) · Zbl 1439.11219
[12] 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) · Zbl 1027.05107
[13] Ehrenborg, R., On posets and Hopf algebras, Adv. Math., 119, 1-25 (1996) · Zbl 0851.16033
[14] Gao, X.; Guo, L., A note on connected cofiltered coalgebras, conilpotent coalgebras and Hopf algebras, Southeast Asian Bull. Math., 43, 313-321 (2019) · Zbl 1449.16058
[15] Geissinger, L., Hopf Algebras of Symmetric Functions and Class Functions, Lecture Notes Math., vol. 579, 168-181 (1977), Springer · Zbl 0366.16002
[16] Gelfand, I.; Krob, D.; Lascoux, A.; Leclerc, B.; Retakh, V.; Thibon, J.-Y., Noncommutative symmetric functions, Adv. Math., 112, 218-348 (1995) · Zbl 0831.05063
[17] Gessel, I., Multipartite P-partitions and inner products of skew Schur functions, Contemp. Math., 34, 289-301 (1984) · Zbl 0562.05007
[18] Guo, L., An Introduction to Rota-Baxter Algebra (2012), International Press · Zbl 1271.16001
[19] Guo, L.; Keigher, W., Baxter algebras and shuffle products, Adv. Math., 150, 117-149 (2000) · Zbl 0947.16013
[20] Guo, L.; Thibon, J.-Y.; Yu, H., Weak quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras, Adv. Math., 344, 1-34 (2019) · Zbl 1403.05160
[21] Hersh, P.; Hsiao, S., Random walks on quasisymmetric functions, Adv. Math., 222, 782-808 (2009) · Zbl 1229.05271
[22] Hivert, F., Hecke algebras, difference operators, and quasi-symmetric functions, Adv. Math., 155, 181-238 (2000) · Zbl 0990.05129
[23] Hoffman, M. E., Quasi-shuffle products, J. Algebraic Comb., 11, 49-68 (2000) · Zbl 0959.16021
[24] Hoffman, M. E.; Ihara, K., Quasi-shuffle products revisited, J. Algebra, 481, 293-326 (2017) · Zbl 1378.16042
[25] Ihara, K.; Kajikawa, J.; Ohno, Y.; Okuda, J., Multiple zeta values vs. multiple zeta-star values, J. Algebra, 332, 187-208 (2011) · Zbl 1266.11093
[26] Joni, S.; Rota, G.-C., Coalgebras and bialgebras in combinatorics, Stud. Appl. Math., 61, 93-139 (1979) · Zbl 0471.05020
[27] Lam, T.; Pylyavskyy, P., Combinatorial Hopf algebras and K-homology of Grassmannians, Int. Math. Res. Not., Article rnm125 pp. (2007) · Zbl 1134.16017
[28] Li, Y., On weak peak quasisymmetric functions, J. Comb. Theory, Ser. A, 158, 449-491 (2018) · Zbl 1391.05260
[29] Luoto, K.; Mykytiuk, S.; van Willigenburg, S., An Introduction to Quasisymmetric Schur Functions. Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux, Springer Briefs in Mathematics (2013), Springer · Zbl 1277.16027
[30] Macdonald, I. G., Symmetric Functions and Hall Polynomials (1995), Oxford University Press: Oxford University Press New York · Zbl 0899.05068
[31] Malvenuto, C., Produits et coproduits des fonctions quasi-symétriques et de l’algèbre des descents (1994), UQAM, Thèse Math
[32] Malvenuto, C.; Reutenauer, C., Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177, 967-982 (1995) · Zbl 0838.05100
[33] Manchon, D., Hopf algebras in renormalisation, (Hazewinkel, M., Handbook of Algebra, Vol. 5 (2008), Elsevier: Elsevier Amsterdam), 365-427 · Zbl 1215.81071
[34] Mantaci, R.; Reutenauer, C., A generalization of Solomon’s algebra for hyperoctahedral groups and other wreath products, Commun. Algebra, 23, 27-56 (1995) · Zbl 0836.20010
[35] Novelli, J.-C.; Thibon, J.-Y., Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions, Discrete Math., 310, 3584-3606 (2010) · Zbl 1231.05278
[36] Petersen, T. K., Enriched P-partitions and peak algebras, Adv. Math., 209, 561-610 (2007) · Zbl 1111.05097
[37] Petersen, T. K., Cyclic descents and P-partitions, J. Algebraic Comb., 22, 343-375 (2005) · Zbl 1107.20011
[38] Reiner, V., Signed posets, J. Comb. Theory, Ser. A, 62, 324-360 (1993) · Zbl 0773.06008
[39] Reiner, V., Signed permutation statistics, Eur. J. Comb., 14, 553-567 (1993) · Zbl 0793.05005
[40] Reutenauer, C., Free Lie Algebras (1993), Oxford University Press: Oxford University Press New York · Zbl 0798.17001
[41] Rota, G.-C., Baxter operators, an introduction, (Kung, J. P.S., Gian-Carlo Rota on Combinatorics, Introductory Papers and Commentaries (1995), Birkhäuser: Birkhäuser Boston) · Zbl 0841.01031
[42] Solomon, L., A Mackey formula in the group ring of a Coxeter group, J. Algebra, 41, 255-264 (1976) · Zbl 0355.20007
[43] Stembridge, J., Enriched P-partitions, Trans. Am. Math. Soc., 349, 763-788 (1997) · Zbl 0863.06005
[44] Stanley, R. P., Enumerative Combinatorics, Vol. 1 (2012), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1247.05003
[45] Stanley, R. P., Ordered structures and partitions, Mem. Am. Math. Soc., 119 (1972) · Zbl 0246.05007
[46] Sweedler, M., Hopf Algebras (1969), Benjamin: Benjamin New York · Zbl 0194.32901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.