====== Category of two-colored pair partitions with neutral blocks ======

The **category of two-colored pair partitions with neutral blocks** is a [[categories of two-colored partitions|category of two-colored partitions]] inducing the co-representation categories of the [[unitary group|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 [(:ref: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 $\Partition{\Pline (1,0) (2,1) \Pline (2,0) (1,1) \Ppoint 0 \Pw:1,2 \Ppoint 1 \Pw:1,2}$. 

===== Associated unitary easy quantum groups =====

The category of two-colored pair partitions with neutral blocks induces the co-representation categories of the [[unitary group|unitary groups]] $(U_N)_{N\in \N}$.

===== References =====


[( :ref:TaWe18 >>
author:  Tarrago, Pierre and Weber, Moritz
title:   The classification of tensor categories of two-colored non-crossing partitions
year:    2018
journal: Journal of Combinatorial Theory, Series A
volume:  154
month:   February
pages:   464--506
url:     https://doi.org/10.1016/j.jcta.2017.09.003
archivePrefix: arXiv
eprint   :1509.00988 
)]