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Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to "half-liberated easy" interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case.

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Compact quantum groups can be studied by investigating their representation categories in analogy to the Schur–Weyl/Tannaka–Krein approach. For the special class of (unitary) “easy” quantum groups, these categories arise from a combinatorial structure: rows of two-colored points form the objects, partitions of two such rows the morphisms. Vertical/horizontal concatenation and reflection give composition, monoidal product and involution. Of the four possible classes $${\mathcal {O}}$$ O , $${\mathcal {B}}$$ B , $${\mathcal {S}}$$ S and $${\mathcal {H}}$$ H of such categories (inspired, respectively, by the classical orthogonal, bistochastic, symmetric and hyperoctahedral groups), we treat the first three—the non-hyperoctahedral ones. We introduce many new examples of such categories. They are defined in terms of subtle combinations of block size, coloring and non-crossing conditions. This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. It is purely combinatorial in nature. The quantum group aspects are left out.