The categories of the hyperoctahedral series are a family of Banica-Speicher categories of partitions, indexed by introduced by Raum and Weber in [RaWe15].

Let be the free product group of many copies of the cyclic group of order and for every let be the image of under the embedding of as the -th free factor in . Then, generates and any group endomorphism of is uniquely determined by its restriction to .

The strong symmetric semigroup is the subsemigroup of the semigroup of group endomorphisms of generated by the endomorphisms defined by for all mappings such that . In other words, is given by all identifications of finitely many letters in words in an alphabet of countably many letters from .

A set is said to be -invariant if for all and .

For every with by the category of the hyperoctahedral series with parameter one denotes the subcategory of the category of all partitions whose morphism class is given by the set of all (uncolored) partitions with the property that the word representation of , if interpreted as a product in (generally the word representation is not a fully reduced word), is an element of the smallest -invariant normal subgroup of which contains and .

Had one allowed in the above definition one would have obtained the category of partitions with blocks of even size. Permitting yields the category of partitions of even size.

The categories of the hyperoctahedral series are special cases of the class of group-theoretical hyperoctahedral categories of partitions.

The category of the hyperoctahedral series with parameter with is the smallest subcategory of containing and the partition [RaWe14], the partition whose word representation is given by .

Via Tannaka-Krein duality for compact quantum groups, the categories of the hyperoctahedral series induce the corepresentation categories of the quantum groups belonging to, as the name suggests, the hyperoctahedral series .