Group-theoretical hyperoctahedral categories of partitions

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

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 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 [RaWe14] algebraically and in [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 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 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 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 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.

References

[RaWe15] Raum, Sven and Weber, Moritz, 2015. Easy quantum groups and quantum subgroups of a semi-direct product quantum group. Journal of Noncommutative Geometry, 9, pp.1261–1293.

[RaWe14] Raum, Sven and Weber, Moritz, 2014. The combinatorics of an algebraic class of easy quantum groups. Infinite Dimensional Analysis, Quantum Probability and related topics, 17.

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Group-theoretical hyperoctahedral categories of partitions

Definition

Canonical Generator

Associated easy quantum group

References

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