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.

在单一易量子群的框架内，我们研究了Brauer的Schur-Weyl方法对正交基团表示理论的类似物。我们考虑具体的组合类别，其形态是由有限集的划分成两个基数的不相交子集形成的；这些集合的点被涂成黑色或白色。这些类别对应于group群和Wang的量子对应物之间的“半解放易”插值。我们完成了所有这些类别的分类，证明了某个自然中途点的子类别与自然数的累加子半群等价；在上一篇文章中已经对中途点以上的类别进行了分类。我们仅使用组合方式来实现这一目标。