Non-group-theoretical hyperoctahedral categories of partitions

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

Definition

A 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 [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 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 even size and has parity-balanced legs. Moreover, if $l\leq 1$ , then it is also 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 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 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 quantum groups.

References

[RaWe16] Raum, Sven and Weber, Moritz, 2016. The full classification of orthogonal easy quantum groups. Communications in Mathematical Physics, 341, pp.751–779.

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Non-group-theoretical hyperoctahedral categories of partitions

Definition

Canonical Generator

Associated easy quantum group

References

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