zbMATH — the first resource for mathematics

Commutative combinatorial Hopf algebras. (English) Zbl 1181.16031
Let \(K\) be a field of characteristic zero, \(R=K[x_{ij}]\) for \(i,j\) bigger than or equal to one, \(J\) the ideal of \(R\) generated by all \(x_{ij}x_{ik}\). For \(f\) an endofunction of \([n]=\{1,2,\dots,n\}\), i.e., \(f\) maps \([n]\) to \([n]\), a homogeneous polynomial \(M_f\) of degree \(n\) in the \(x_{ij}\) is defined. The \(M_f\), as \(n\) and \(f\) vary, span a subalgebra EQSym of \(R/J\).
The product \(M_fM_g\) is a linear combination of the \(M_h\) with integral coefficients defined in terms of shifted concatenations, which leads to a product of \(f\) and \(g\). This in turn leads to a comultiplication on each \(M_h\), so that EQSym is a commutative noncocommutative graded Hopf algebra. Its graded dual ESym is free over those elements \(S^f\) of the dual basis to the \(M_f\) for which \(f\) is connected, i.e., \(f\) is not a nontrivial shifted concatenation. If one only uses the \(M_s\) for \(s\) a permutation of some \([n]\), then one gets a Hopf subalgebra SQSym of EQSym, whose graded dual SSym is free over the \(S_s\) for \(s\) connected.
It turns out that SSym is isomorphic to the Grossman-Larson Hopf algebra of heap ordered trees [R. Grossman and R. G. Larson, J. Algebra 126, No. 1, 184-210 (1989; Zbl 0717.16029)] and also to the Hopf algebra of permutations of F. Patras and C. Reutenauer [Mosc. Math. J. 4, No. 1, 199-216 (2004; Zbl 1103.16026)]. Some subalgebras of SQSym are discussed, including symmetric functions in noncommuting variables (dual), quasi-symmetric functions, and ordinary symmetric functions. SQSym is noncommutative as an algebra. It can be realized as a Hopf algebra within \(K\langle a_{ij}\rangle\), polynomials in noncommuting \(a_{ij}\) for \(i,j\) bigger than or equal to one, whose operations can be described in terms of the cycle structure of permutations. The paper concludes with some attempts to relate to the spirit of quantum groups.

16T30 Connections of Hopf algebras with combinatorics
05E05 Symmetric functions and generalizations
05C05 Trees
Full Text: DOI
[1] Aguiar, M.; Sottile, F., Cocommutative Hopf algebras of permutations and trees, Journal of Algebraic Combinatorics, 22, 451-470, (2005) · Zbl 1094.16024
[2] Bergeron, F., Labelle, G., & Leroux, P. (1998). Combinatorial species and tree-like structures. Encyclopedia of mathematics and its applications, Vol. 67. Cambridge: Cambridge University Press. · Zbl 0888.05001
[3] Bergeron, N., Reutenauer, C., Rosas, M., & Zabrocki, M. (2005). Invariants and coinvariants of the symmetric group in noncommuting variables. Preprint ArXiv:math.CO/0502082. · Zbl 1180.16025
[4] Brouder, C.; Frabetti, A., Renormalization of QED with planar binary trees, European Journal of Physics C, 19, 715-741, (2001) · Zbl 1099.81568
[5] Brouder, C.; Oeckl, R., Quantum groups and quantum field theory: I. The free scalar field, 63-90, (2004), Hauppauge
[6] Connes, A.; Kreimer, D., Hopf algebras, renormalization and noncommutative geometry, Communications in Mathematical Physics, 199, 203-242, (1998) · Zbl 0932.16038
[7] Duchamp, G.; Hivert, F.; Thibon, J.-Y., Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, International Journal of Algebra and Computation, 12, 671-717, (2002) · Zbl 1027.05107
[8] Gelfand, I. M.; Krob, D.; Lascoux, A.; Leclerc, B.; Retakh, V. S.; Thibon, J.-Y., Noncommutative symmetric functions, Advances in Mathematics, 112, 218-348, (1995) · Zbl 0831.05063
[9] Grossman, R.; Larson, R. G., Hopf-algebraic structure of families of trees, J. Algebra, 126, 184-210, (1989) · Zbl 0717.16029
[10] Hivert, F., Hecke algebras, difference operators, and quasi-symmetric functions, Advances in Mathematics, 155, 181-238, (2000) · Zbl 0990.05129
[11] Hivert, F.; Novelli, J.-C.; Thibon, J.-Y., The algebra of binary search trees, Theoretical Computer Science, 339, 129-165, (2005) · Zbl 1072.05052
[12] Hivert, F., Novelli, J.-C., & Thibon, J.-Y. (2005). Commutative Hopf algebras of permutations and trees. Preprint ArXiv:math.CO/0502456. · Zbl 1072.05052
[13] Joyal, A. (1981). Une théorie combinatoire des séries formelles. Advances in Mathematics, 42, 1-82. · Zbl 0491.05007
[14] Krob, D.; Thibon, J.-Y., Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at \(q=0,\) Journal of Algebraic Combinatorics, 6, 339-376, (1997) · Zbl 0881.05120
[15] Leclerc, B.; Scharf, T.; Thibon, J.-Y., Noncommutative cyclic characters of symmetric groups, Journal of Combinatorial Theory Series A, 75, 55-69, (1996) · Zbl 0857.20005
[16] Li, L.; Zhang, P., Twisted Hopf algebras, Ringel-Hall algebras, and Green’s categories, Journal of Algebra, 231, 713-743, (2000) · Zbl 0995.16031
[17] Loday, J.-L.; Ronco, M. O., Hopf algebra of the planar binary trees, Advances in Mathematics, 139, 293-309, (1998) · Zbl 0926.16032
[18] Macdonald, I. G. (1995). Symmetric functions and Hall polynomials (2nd ed). Oxford: Oxford University Press. · Zbl 0824.05059
[19] Malvenuto, C.; Reutenauer, C., Duality between quasi-symmetric functions and the Solomon descent algebra, Journal of Algebra, 177, 967-982, (1995) · Zbl 0838.05100
[20] Novelli, J.-C., & Thibon, J.-Y. (2006). Construction de trigèbres dendriformes. Comptes Rendus. Académie des Sciences Paris Série I, 342, 365-369. · Zbl 1101.17003
[21] Novelli, J.-C.; Thibon, J.-Y., Hopf algebras and dendriform structures arising from parking functions, Fundamenta Mathematicae, 193, 189-241, (2007) · Zbl 1127.16033
[22] Novelli, J.-C.; Thibon, J.-Y., Parking functions and descent algebras, Annals of Combinatorics, 11, 59-68, (2007) · Zbl 1115.05095
[23] Novelli, J.-C., Thibon, J.-Y., & Thiéry, N. M. (2004). Algèbres de Hopf de graphes. Comptes Rendus. Académie des Sciences Paris Série A, 339(9), 607-610. · Zbl 1062.16046
[24] Novelli, J.-C., & Thibon, J.-Y. (2005). Noncommutative symmetric functions and Lagrange inversion. Preprint ArXiv:math.CO/0512570.
[25] Patras, F.; Reutenauer, C., On descent algebras and twisted bialgebras, Moscow Mathematical Journal, 4, 199-216, (2004) · Zbl 1103.16026
[26] Poirier, S., & Reutenauer, C. (1995). Algèbre de Hopf des tableaux. Annales des Science Mathématiques du Québec, 19, 79-90. · Zbl 0835.16035
[27] Reutenauer, C. (1993). Free Lie algebras, London Mathematical Society Monographs. New York: Oxford University Press. · Zbl 0798.17001
[28] Rey, M. (2007). Algebraic construction on set partitions In Proc. FPSAC’07, Nankai University, July 2007.
[29] Sagan, B.; Rosas, M., Symmetric functions in noncommuting variables, Transactions of the American Mathematical Society, 358, 215-232, (2006) · Zbl 1071.05073
[30] Scharf, T.; Thibon, J.-Y., A Hopf-algebra approach to inner plethysm, Advances in Mathematics, 104, 30-58, (1994) · Zbl 0830.20027
[31] Sloane, N. J. A. The on-line encyclopedia of integer sequences. http://www.research.att.com/ njas/sequences/. · Zbl 1044.11108
[32] Thibon, J.-Y.; Ung, B. C. V., Quantum quasi-symmetric functions and Hecke algebras, Journal of Physics A, 29, 7337-7348, (1996) · Zbl 0962.05060
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.