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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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