====== Non-group-theoretical hyperoctahedral categories of partitions ======

The **non-group-theoretical hyperoctahedral categories** are a one-parameter family of [[category_of_partitions|Banica-Speicher categories of partitions]], indexed by the natural numbers with infinity, introduced by Raum and Weber in [(:ref:RaWe16)].

===== 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 **non-group-theoretical** if $\Paabaab\notin \Cscr$. If $\Cscr$ has both these properties, we call it **non-group-theoretical hyperoctahedral**. The individual categories which belong to this class have no commonly used proper names, 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 non-group-theoretical hyperoctahedral categories in [(:ref:RaWe16)]. There is a bijection between the class of all such categories and the set $\N\cup\{\infty\}$. 

For every $l\in \N\cup\{\infty\}$ a partition $p\in\Pscr$ is said to belong to the set of morphisms of the **non-group-theoretical hyperoctahedral category with parameter** $l$ if the following conditions are met:
  * $p$ has //blocks of even size//,  meaning $|B|\in 2\N$ for every block $B$ of $p$ (see [[category of partitions with blocks of even size]]) .
  * $p$ has the property that for all blocks $B$ and $B'$ of $p$ with $B\neq B'$ and all legs $i,j\in B$ with $i\neq j$ the set $[i,j]_p\cap B'$ has even cardinality (which includes $0$), where $[i,j]=\{ k\,\vert\, i\preceq k\preceq j\}$. In other words, between any two legs of one block any other block may only have evenly many legs.
  * $p$ satisfies $\mathrm{wdepth}(p)\leq l$, which means that $p$ contains no W of depth larger than $l$ if $l\in\N$ and which is a vacuous condition if $l=\infty$. For every $k\in\N$ we say that $p$ **contains a W of depth** $k$ if there exist letters $a_1,\ldots,a_k$ and words $X_1^\alpha,\ldots, X_k^\alpha$, $X_1^\beta,\ldots,X_{k-1}^\beta$, $X_1^\gamma,\ldots, X_k^\gamma$, $X_1^\delta,\ldots, X_{k-1}^\delta$ and $Y_1$, $Y_2$, $Y_3$ such that 
    * the [[partition#word_representation|word representation]] of $p$ is given by $p=Y_1 S_\alpha X_k^\alpha S_\beta Y_2 S_\gamma X_k^\gamma S_\delta Y_3,$ 
    * where $S_\alpha=a_1 X_1^\alpha a_2 X_2^\alpha\ldots a_{k-1}X_{k-1}^\alpha a_k$, 
    * where $S_\beta=a_kX_{k-1}^\beta a_{k-1}X_{k-2}^\beta\ldots a_2 X_1^\beta a_1$, 
    * where $S_\gamma=a_1 X_1^\gamma a_2 X_2^\gamma\ldots a_{k-1}X_{k-1}^\gamma a_k$,
    * where $S_\delta=a_k X_{k-1}^\delta a_{k-1}X_{k-2}^\delta\ldots a_2 X_1^\delta a_1$ and 
    * where for every $i=1,\ldots,k$ the letter $a_i$ appears and //odd// number of times in each word $S_\alpha$, $S_\beta$, $S_\gamma$ and $S_\delta$ 
    * where $Y_1$, $Y_2$ and $Y_3$ contain none of the letters $a_1,\ldots,a_k$.

Any such partition is necessarily of [[category of partitions of even size|even size]] and has [[category of partitions with blocks of even size and even distances between legs|parity-balanced legs]]. Moreover, if $l\leq 1$, then it is also [[category of all non-crossing partitions|non-crossing]].

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

For every $l\in \N$ the non-group-theoretical hyperoctahedral category with parameter $l$ is the subcategory of $\Pscr$ generated by the partition $\pi_l\in \pscr(0,4l)$ whose [[partition#word_representation|word representation]] is
$$\pi_l=\mathsf{a}_1\mathsf{a}_2\cdots\mathsf{a}_l\mathsf{a}_l\cdots\mathsf{a}_2\mathsf{a}_1\mathsf{a}_1\mathsf{a}_2\cdots\mathsf{a}_l\mathsf{a}_l\cdots\mathsf{a}_2\mathsf{a}_1.$$

The non-group-theoretical hyperoctahedral category with parameter $\infty$ is not finitely generated. It is the smallest subcategory of $\Pscr$ containing the set $\{\pi_k\vert k\in\N\}$ of generators.

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

Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, for every $l\in\N\cup\{\infty\}$ the non-group-theoretical hyperoctahedral category with parameter $l$ corresponds to a family of [[non-group-theoretical_hyperoctahedral_easy_orthogonal_quantum_groups|non-group-theoretical hyperoctahedral easy 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
)]