Category of two-colored pair partitions with neutral blocks

The category of two-colored pair partitions with neutral blocks is a category of two-colored partitions inducing the co-representation categories of the unitary groups.

Definition

By the category of two-colored pair partitions with neutral blocks one denotes the subcategory of the category of all two-colored partitions $\Pscr^{\circ\bullet}$ whose morphism class is the set of all pair partitions with neutral blocks. It was introduced by Tarrago and Weber in [TaWe18], Theorem 8.3 under the name $\mathcal{O}_{\mathrm{grp},\mathrm{loc}}$ .

A two-colored partition $p\in\Pscr^{\circ\bullet}$ is called a pair partition (see category of all pair partitions in the uncolored case), if every block $B$ of $p$ satisfies $|B|=2$ .
$p$ is said to have neutral blocks if every block $B$ of $p$ has vanishing color sum $\sigma_p(B)=0$ . In other words, the numbers of, on the one hand, upper $\bullet$ -colored plus lower $\circ$ -colored legs of $B$ and, on the other hand, upper $\circ$ -colored plus lower $\bullet$ -colored legs of $B$ coincide.
The name set of all pair partitions partitions with neutral blocks is to be taken literally.

The set of two-colored pair partitions with neutral blocks is the morphism set of the subcategory of $\Pscr^{\circ\bullet}$ generated by the two-colored partition $\Partition{\Pline (1,0) (2,1) \Pline (2,0) (1,1) \Ppoint 0 \Pw:1,2 \Ppoint 1 \Pw:1,2}$ .