Séminaire Lotharingien de Combinatoire, B61Aj (2010), 22 pp.

Nantel Bergeron and Philippe Choquette

Hyperoctahedral Species

Abstract. We introduce hyperoctahedral species ($\H$-species) or species of type B, which are analogous to the classical tensor species, but on which we consider the action of the groups of signed permutations. We give a bistrong monoidal functor, a functor which preserves Hopf monoids, between the monoidal categories of species and H-species. We also define bilax monoidal functors (functors which preserve the structure of bimonoids) between the category of H-species and the category of graded vector spaces. Using these functors, the combinatorial Hopf algebra DQSym is shown to arise from the cofree comonoid on the exponential species.

Received: December 10, 2008. Revised: September 15, 2009; September 16, 2010. Accepted: September 29, 2010.

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