The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree

Abstract

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of looking for bases which generate the space we define orders on the set partitions and set compositions which allow us to define bases which have simple and natural rules for the product of basis elements.

22 pages; 9 figures in eps form

References in corpus (1)

• Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

Cited by in corpus (13)

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• Skew quasisymmetric Schur functions and noncommutative Schur functions
• Antipode and Primitive elements in the Hopf Monoid of Super Characters
• A supercharacter table decomposition via power-sum symmetric functions
• A comodule-bialgebra structure for word-series substitution and mould composition
• Scheduling Problems
• Strong forms of linearization for Hopf monoids in species
• Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions
• On the Hadamard product of Hopf monoids
• Set partitions and integrable hierarchies
• Free decomposition spaces
• Symmetric functions in noncommuting variables in superspace