Coloured peak algebras and Hopf algebras

Abstract

For $G$ a finite abelian group, we study the properties of general equivalence relations on $G_n=G^n\rtimes \SG_n$, the wreath product of $G$ with the symmetric group $\SG_n$, also known as the $G$-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of $\k G_n$ as well as graded connected Hopf subalgebras of $\bigoplus_{n\ge o} \k G_n$. In particular we construct a $G$-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or $G$-coloured descent algebra). We show that the direct sum of the $G$-coloured peak algebras is a Hopf algebra. We also have similar results for a $G$-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the $G$-coloured descent Hopf algebra whose image is the $G$-coloured peak Hopf algebra. We outline a theory of combinatorial $G$-coloured Hopf algebra for which the $G$-coloured quasi-symmetric Hopf algebra and the graded dual to the $G$-coloured peak Hopf algebra are central objects.

26 pages latex2e

References in corpus (5)

• The peak algebra and the descent algebras of types B and D
• The peak algebra and the Hecke-Clifford algebras at $q=0$
• Free quasi-symmetric functions of arbitrary level
• Representation theory of the 0-Ariki-Koike-Shoji algebras
• The Algebra of Binary Search Trees

Cited by in corpus (14)

• Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras
• Combinatorial Hopf algebras
• Cambrian Hopf Algebras
• Peaks Sets of Classical Coxeter Groups
• Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables
• Specializations of colored quasisymmetric functions and Euler-Mahonian identities
• A semigroup approach to wreath-product extensions of Solomon's descent algebras
• Extending the descent-to-peak map and its applications
• Theta maps for combinatorial Hopf algebras
• Enriched $P$-partitions and peak algebras (extended abstract)
• Colored posets and colored quasisymmetric functions
• A generalization of the dual immaculate quasisymmetric functions in partially commutative variables
• Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras
• Enriched $P$-partitions and peak algebras