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

Wenhua Zhao
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Abstract

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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Wenhua Zhao. 2007-10-28. A Noncommutative Symmetric System over the Grossman-Larson Hopf Algebra of Labeled Rooted Trees. https://doi.org/10.1007/s10801-007-0100-5

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