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Structure of the Malvenuto-Reutenauer Hopf algebra of permutations. (English) Zbl 1056.05139
Summary: We analyze the structure of the Malvenuto-Reutenauer Hopf algebra \({\mathfrak S}\)Sym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. In addition, we describe the structure constants of the multiplication as a certain number of facets of the permutahedron. As a consequence we obtain a new interpretation of the product of monomial quasi-symmetric functions in terms of the facial structure of the cube. The Hopf algebra of Malvenuto and Reutenauer has a linear basis indexed by permutations. Our results are obtained from a combinatorial description of the Hopf algebraic structure with respect to a new basis for this algebra, related to the original one via Möbius inversion on the weak order on the symmetric groups. This is in analogy with the relationship between the monomial and fundamental bases of the algebra of quasi-symmetric functions. Our results reveal a close relationship between the structure of the Malvenuto-Reutenauer Hopf algebra and the weak order on the symmetric groups.

MSC:
05E05 Symmetric functions and generalizations
06A11 Algebraic aspects of posets
05E15 Combinatorial aspects of groups and algebras (MSC2010)
06A07 Combinatorics of partially ordered sets
06A15 Galois correspondences, closure operators (in relation to ordered sets)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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[1] Aguiar, M.; Santos, W.F., Galois connections for incidence Hopf algebras of partially ordered sets, Adv. in math, 151, 1, 71-100, (2000), MR2001f:06007 · Zbl 0945.06002
[2] Billera, L.J.; Liu, N., Noncommutative enumeration in graded posets, J. algebraic combin, 12, 1, 7-24, (2000), MR2001h:05009 · Zbl 0971.05005
[3] A. Björner, Orderings of Coxeter groups, Combinatorics and algebra (Boulder, Co., 1983), American Mathematical Society, Providence, RI, 1984, pp. 175-195, MR86i:05024.
[4] Björner, A.; Vergnas, M.L.; Sturmfels, B.; White, N.; Ziegler, G.M., Oriented matroids, (1999), Cambridge University Press Cambridge, MR2000j:52016
[5] Blattner, R.J.; Cohen, M.; Montgomery, S., Crossed products and inner actions of Hopf algebras, Trans. amer. math. soc, 298, 2, 671-711, (1986), MR87k:16012 · Zbl 0619.16004
[6] L. Comtet, The art of finite and infinite expansions, Advanced Combinatorics, Enlarged Edition, D. Reidel Publishing Co., Dordrecht, 1974, MR57#124.
[7] G. Duchamp, F. Hivert, J.-Y. Thibon, Some generalizations of quasi-symmetric functions and noncommutative symmetric functions, Formal Power Series and Algebraic Combinatorics (Moscow, 2000), Springer, Berlin, 2000, pp. 170-178, MR2001k:05202.
[8] Duchamp, G.; Hivert, F.; Thibon, J., Noncommutative symmetric functions vifree quasi-symmetric functions and related algebras, Internat. J. algebra comput, 12, 5, 671-717, (2002), MR1935570 · Zbl 1027.05107
[9] P. Edeleman, Geometry and the Möbius function of the weak Bruhat order of the symmetric group, 1983 preprint.
[10] Ehrenborg, R., On posets and Hopf algebras, Adv. in math, 119, 1, 1-25, (1996), MR97e:16079 · Zbl 0851.16033
[11] Foata, D.; Schützenberger, M.-P., Major index and inversion number of permutations, Math. nachr, 83, 143-159, (1978), MR81d:05007 · Zbl 0319.05002
[12] Gelfand, I.M.; A. Lascoux, D.Krob; Leclerc, B.; Retakh, V.S.; Thibon, J.-Y., Noncommutative symmetric functions, Adv. in math, 112, 2, 218-348, (1995), MR96e:05175 · Zbl 0831.05063
[13] I.M. Gessel, Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Co., 1983) American Mathematical Society, Providence, RI, 1984, pp. 289-317. MR86k:05007.
[14] Guilbaud, G.Th.; Rosenstiehl, P., Analyse algébrique d’un scrutin, M. sci. humaines, 4, 9-33, (1960)
[15] Hoffman, M.E., Quasi-shuffle products, J. algebraic combin, 11, 1, 49-68, (2000), MR2001f:05157 · Zbl 0959.16021
[16] Jöllenbeck, A., Nichtkommutative charaktertheorie der symmetrischen gruppen, [noncommutative theory of characters of symmetric groups], Bayreuth. math. schr, 56, 1-41, (1999) · Zbl 0931.20013
[17] A. Lentin, Équations dans les monoı̈des libres, Mathématiques et Sciences de l’Homme, Vol. 16. Mouton, Gauthier-Villars, Paris, 1972, iv+160pp. · Zbl 0258.20058
[18] Loday, J.-L., Homotopical syzygies, une dégustation topologique [topological morsels]homotopy theory in the swiss alps (arolla, 1999), Contemp. math, 265, 99-127, (2000), MR2001k:55022
[19] Loday, J.-L.; Ronco, M.O., Hopf algebra of the planar binary trees, Adv. in math, 139, 2, 293-309, (1998), MR99m:16063 · Zbl 0926.16032
[20] Loday, J.-L.; Ronco, M.O., Order structure on the algebra of permutations and of planar binary trees, J. algebraic combin, 15, 3, 253-270, (2002), MR1900627 · Zbl 0998.05013
[21] I.G. Macdonald, Symmetric functions and Hall Polynomials, 2nd Edition, The Clarendon Press Oxford University Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications, MR96h:05207. · Zbl 0824.05059
[22] C. Malvenuto, Produits et coproduits des fonctions quasi-symétriques et de l’algèbre des descents, Vol. 16, Laboratoire de combinatoire et d’informatique mathématique (LACIM), Univ. du Québec à Montréal, Montréal, 1994.
[23] Malvenuto, C.; Reutenauer, C., Duality between quasi-symmetric functions and the Solomon descent algebra, J. algebra, 177, 3, 967-982, (1995), MR97d:05277 · Zbl 0838.05100
[24] Milgram, R.J., Iterated loop spaces, Ann. of math. (2), 84, 386-403, (1966), MR34 #6767 · Zbl 0145.19901
[25] Milnor, J.W.; Moore, J.C., On the structure of Hopf algebras, Ann. of math. (2), 81, 211-264, (1965), MR30 #4259 · Zbl 0163.28202
[26] S. Montgomery, Hopf algebras and their actions on rings, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1993. MR94i:16019. · Zbl 0793.16029
[27] Patras, F.; Reutenauer, C., Lie representations and an algebra containing Solomon’s, J. algebraic combin, 16, 3, 301-314, (2002), (2003) · Zbl 1056.16032
[28] Poirier, S.; Reutenauer, C., Algèbres de Hopf de tableaux, Ann. sci. math. Québec, 19, 1, 79-90, (1995), MR96g:05146 · Zbl 0835.16035
[29] Reutenauer, C., Free Lie algebras, (1993), The Clarendon Press, Oxford University Press New York, Oxford Science Publications. MR94j:17002 · Zbl 0798.17001
[30] Reprinted in: J.P.S. Kung (Ed.), Gian-Carlo Rota on Combinatorics: Introductory papers and commentaries, Birkhäuser, Boston, 1995, MR30 #4688. · Zbl 0121.02406
[31] N.J.A. Sloane, An on-line version of the encyclopedia of integer sequences, Electron. J. Combin. 1 (1994) Feature 1, approx. 5 pp. (electronic), , MR95b:05001. · Zbl 0815.11001
[32] R.P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original, MR98a:05001. · Zbl 0889.05001
[33] R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin, MR2000k:05026. · Zbl 0928.05001
[34] Stembridge, J.R., Enriched P-partitions, Trans. amer. math. soc, 349, 2, 763-788, (1997), MR97f:06006 · Zbl 0863.06005
[35] Takeuchi, M., Free Hopf algebras generated by coalgebras, J. math. soc. Japan, 23, 561-582, (1971) · Zbl 0217.05902
[36] Thibon, J.-Y.; Ung, B.-C.-V., Quantum quasi-symmetric functions and Hecke algebras, J. phys. A, 29, 22, 7337-7348, (1996), MR97k:05204 · Zbl 0962.05060
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