Coloured peak algebras and Hopf algebras.

Bergeron, Nantel ; Hohlweg, Christophe

Journal of Algebraic Combinatorics, Tome 24 (2006) no. 3, pp. 299-330.

Résumé

Summary: For $G$ a finite abelian group, we study the properties of general equivalence relations on $G _{ n } = G ^{ n }\? \mathfrak S {\mathfrak S} _{ n }$, the wreath product of $G$ with the symmetric group $\mathfrak S {\mathfrak S} _{ n }$, also known as the $G$-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of $\Bbbk $ Bbbk $G _{ n }$ as well as graded connected Hopf subalgebras of ? $_{ n\geq o } \Bbbk $ Bbbk $G _{ n }$. 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.

Détail

Classification : 16S99, 05E05, 05E10, 16S34, 16W30, 20B30
Keywords: keywords bialgebra, combinatorial Hopf algebra, wreath product, Coxeter group, symmetric group, hyperoctahedral group, peak algebra, planar binary tree, descent algebra