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Home > Vol 24, No 3 (2006)

Nantel Bergeron¹ and Christophe Hohlweg²

¹York University Department of Mathematics and Statistics Toronto Ontario M3J 1P3 Canada
²The Fields Institute 222 College Street Toronto Ontario Canada M5T 3J1

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.

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