====== Category of partitions with blocks of even size ======

The **category of partitions with blocks of even size** is a [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[wp>hyperoctahedral group|hyperoctahedral groups]]. 

===== Definition =====
By the **category of partitions with blocks of even size** one denotes the subcategory of the [[category of all partitions]] $\Pscr$ whose morphism set is the //set of all partitions with blocks of even size//. It was introduced by Banica, Bichon and Collins in [(:ref:BanBichColl07)]. 

This name is to be taken literally. A partition  $p\in\Pscr$ is said to have **blocks of even size** if every block of $p$ has an even number of legs.

===== Canonical generator =====

The category of partitions with blocks of even size is the subcategory of $\Pscr$ generated by the set $\{\crosspart,\fourpart\}$ of partitions. 

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


Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the category of all partitions with blocks of even size corresponds to the family $(H_N)_{N\in \N}$ of [[wp>hyperoctahedral group|hyperoctahedral groups]].

===== References =====

[( :ref:BanSp09 >>
author:  Banica, Teodor and Speicher, Roland
title:   Liberation of orthogonal Lie groups
year:    2009
journal: Advances in Mathematics
volume:  222
issue:   4
pages:   1461--150
url:     https://doi.org/10.1016/j.aim.2009.06.009
archivePrefix: arXiv
eprint   :0808.2628
)]

[( :ref:BanBichColl07 >>
author   : Banica Teodor and Bichon Julien and Collins Benoit
title    : The hyperoctahedral quantum group
journal  : Journal of the Ramanujan Mathematical Society
year     : 2007
volume   : 22
number   : 4
pages    : 345--384
archivePrefix: arXiv
eprint   :0701859
)]