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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, i.e. every block of $p$ has even number of legs.
$p$ 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 (see category of partitions of even size) and has even distances between blocks (see category of partitions with blocks of even size and even distances between legs ). Moreover, if $l\leq 1$ , then it is also non-crossing (see category of all non-crossing partitions.)

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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