====== Categories of the higher hyperoctahedral series ======

The **categories of the higher hyperoctahedral series** are a family of [[category_of_partitions|Banica-Speicher categories of partitions]], indexed by $\{3,4,\ldots,\infty\}$ introduced by Raum and Weber in [(:ref:RaWe15)].

===== Definition =====
Let $\Z_2^{\ast\infty}$ be the [[wp>free product]] group of $\aleph_0$ many copies of the cyclic group $\Z_2\equiv \Z/2\Z$ of order $2$ and for every $k\in\N$ let $a_k$ be the image of $1\equiv 1+2\Z$ under the embedding of $\Z_2$ as the $k$-th free factor in $\Z_2^{\ast\infty}$. Then, $\{a_k\,\vert\,k\in\N\}$ generates $\Z_2^{\ast\infty}$ and any group endomorphism of $\Z_2^{\ast\infty}$ is uniquely determined by its restriction to $\{a_k\,\vert\, k\in\N\}$. 

The **strong symmetric semigroup** $\mathrm{sS}_\infty$ is the subsemigroup of the semigroup $\mathrm{End}(\Z_2^{\ast\infty})$ of group endomorphisms of $\Z_2^{\ast\infty}$ generated by the endomorphisms defined by $a_k\mapsto a_{i(k)}$ for all mappings $i:\N\to\N$ such that $|\N\backslash i(\N)|<\infty$. In other words, $\mathrm{sS}_\infty$ is given by all identifications of finitely many letters in words in an alphabet of countably many letters from $\Z_2$. 

A set $A\subseteq \Z_2^\infty$ is said to be $\mathrm{sS}_\infty$-**invariant** if $\varphi(w)\in A$ for all $w\in A$ and $\varphi\in \mathrm{sS}_\infty$.

For every $s\in \N\cup \{\infty\}$ with $3\leq s$ by the **category of the higher hyperoctahedral series with parameter** $s$ one denotes the subcategory of the [[category of all partitions]] $\Pscr$ whose morphism class is given by the set of all (uncolored) partitions $p\in\Pscr$ with the property that the [[partition#word_representation|word representation]] of $p$, if interpreted as a product in $\Z_2^{\ast\infty}$ (generally the word representation is not a fully reduced word), is an element of,
  * if $s=\infty$, the trivial subgroup of $\Z^{\ast\infty}_2$,
  * if $s<\infty$, the smallest $\mathrm{sS}_\infty$-invariant normal subgroup of $\Z^{\ast\infty}_2$ which contains $(a_1a_2)^s$.

Had one allowed $s=2$ in the above definition one would have obtained the [[category of partitions with blocks of even size]]. Permitting $s=1$ yields the [[category of partitions of even size]].

The categories of the higher hyperoctahedral series are special cases of the class of [[group-theoretical hyperoctahedral categories of partitions]].

===== Canonical Generator =====

The category of the higher hyperoctahedral series with parameter $s\in N\cup\{\infty\}$ with $3\leq s$ is the smallest subcategory of $\Pscr$ containing,
  * if $s=\infty$, the partition $\Paabaab$,
  * if $s<\infty$,  the partition $h_s$  [(:ref:RaWe14)], the partition whose [[partition#word_representation|word representation]] is given by $(\mathsf{ab})^s$. 


===== Associated easy quantum group =====

Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the categories of the higher hyperoctahedral series induce the corepresentation categories of the quantum groups belonging to, as the name suggests, the [[higher hyperoctahedral series]] $(H^{[s]}_N)_{N\in \N,s\in \{3,\ldots,\infty\}}$.


===== References =====

[( :ref:RaWe16 >>
author:  Raum, Sven and Weber, Moritz 
title:   The full classification of orthogonal easy quantum groups
year:    2016
journal: Communications in Mathematical Physics
volume:  341
issue:   3
pages:   751--779
url:     https://doi.org/10.1007/s00220-015-2537-z
archivePrefix: arXiv
eprint   :1312.3857
)]


[( :ref:RaWe15 >>
author:  Raum, Sven and Weber, Moritz 
title:   Easy quantum groups and quantum subgroups of a semi-direct product quantum group
year:    2015
journal: Journal of Noncommutative Geometry
volume:  9
issue:   4
pages:   1261--1293
url:     https://doi.org/10.4171/JNCG/223 
archivePrefix: arXiv
eprint   :1311.7630v2
)]

[( :ref:RaWe14 >>
author:  Raum, Sven and Weber, Moritz 
title:   The combinatorics of an algebraic class of easy quantum groups
year:    2014
journal: Infinite Dimensional Analysis, Quantum Probability and related topics
volume:  17
issue:   3
url:     https://doi.org/10.1142/S0219025714500167
archivePrefix: arXiv
eprint   :1312.1497v1
)]