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====== Group-theoretical hyperoctahedral categories of partitions ======

The **group-theoretical hyperoctahedral categories** are a family of [[category_of_partitions|Banica-Speicher categories of partitions]], indexed by the set of strongly symmetric reflection groups on countably many generators, introduced by Raum and Weber in [(:ref:RaWe15)].

===== Definition =====

A [[category_of_partitions|category of (uncolored) partitions]] $\Cscr\subseteq \Pscr$ is called **hyperoctahedral** if $\fourpart\in \Cscr$ and $\singleton\otimes\singleton\notin \Cscr$. It is said to be **group-theoretical** if $\Paabaab\in \Cscr$. If $\Cscr$ has both these properties, we call it **group-theoretical hyperoctahedral**. Only two subfamilies of this class have commonly used proper names (see [[categories of the hyperoctahedral series]] and [[categories of the higher hyperoctahedral series]]), which is why they are addressed by their classification according to the group-theoretical/non-group-theoretical and hyperoctahedral/non-hyperoctahedral distinctions. 

Raum and Weber determined all __group-theoretical__ (hyperoctahedral as well as non-hyperoctahedral) categories in [(:ref:RaWe14)] algebraically and in [(:ref:RaWe15)] by purely combinatorial means. There is a bijection between the class of all such categories and the set of //strongly symmetric reflection groups on countably many generators//: 
  * 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 **strongly symmetric reflection group (on countably many generators)** is now a quotient group $\Z_2^{\ast\infty}/A$ of  $\Z_2^{\ast \infty}$ by a normal subgroup $A$ which is invariant under the action of $\mathrm{sS}_\infty$, the latter condition meaning $\varphi(w)\in A$ for all $w\in A$ and $\varphi\in \mathrm{sS}_\infty$.

For every strongly symmetric reflection group $\Z^{\ast\infty}_2/A$ a partition $p\in\Pscr$ is said to belong to the set of morphisms of the **group-theoretical category with parameter** $A$ if the [[partition#word_representation|word representation]] of $p$, if interpreted as a product in $\Z_2^\infty$ (generally the word representation is not a fully reduced word), is an element of $A$.

The __group-theoretical hyperoctahedral__ categories of (uncolored) partitions are now precisely the group-theoretical categories corresponding to reflection groups other than $\Z_2$ and the trivial group $\{e\}$. In other words, one needs to exclude the values $A=\langle a_1a_2\rangle$ and $A=\langle a_1 \rangle$ as admissible $\mathrm{sS}_\infty$-invariant normal subgroups of $\Z_2^{\ast\infty}$. Those two correspond to, respectively, the [[category of partitions of even size]] and the [[category of all partitions]], which are the only non-hyperoctahedral group-theoretical categories.

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

The [[partition#word_representation|word representation]] of a partition associates with that partition an element of $\Z_2^{\ast\infty}$. Conversely, every element of $\Z_2^{\ast\infty}$, i.e., reduced word, can be interpreted as the word representation of a unique partition up to rotations. For every strongly symmetric reflection group $\Z^{\ast\infty}_2/A$ and every generator $G\subseteq A$ of $A$ in $\Z_2^{\ast\infty}$, the group-theoretical hyperoctahedral category with parameter $A$ is generated by the set of partitions whose word representations are elements of $A$.

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

Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, for every strongly symmetric reflection group $A$ the group-theoretical hyperoctahedral category with parameter $A$ corresponds to a family of [[group-theoretical_hyperoctahedral_easy_orthogonal_quantum_groups|group-theoretical hyperoctahedral easy orthogonal quantum groups]].

===== 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
)]
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