A Noncommutative Symmetric System over the Grossman-Larson Hopf Algebra of Labeled Rooted Trees

Details A Noncommutative Symmetric System over the Grossman-Larson Hopf Algebra of Labeled Rooted Trees A Noncommutative Symmetric System over the Grossman-Larson Hopf Algebra of Labeled Rooted Trees

2005-09-07

arxiv.org/abs/math/0509136v3

J. Alg. Comb. 28 (2008), 235--260.

Mathematics

Combinatorics

Latex, 30 pages. Following the referees' suggestions, several places have been improved. In particular, some diagrams of roote

Scientific paper

10.1007/s10801-007-0100-5

In this paper, we construct explicitly a noncommutative symmetric (${\mathcal N}$CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the ${\mathcal N}$CS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficients of the third generating function of the constructed ${\mathcal N}$CS system with the order polynomials of rooted trees is also given and proved.

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