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category_of_partitions_of_even_size

Category of partitions of even size

The category of partitions of even size is a Banica-Speicher category of partitions inducing the corepresentation category of the modified symmetric groups.

Definition

By the category of partitions of even size one denotes the subcategory of the category of all partitions $\Pscr$ whose morphism class is the set of all partitions of even size. It was introduced by Banica and Speicher in [BanSp09].

For all $k,l\in \{0\}\cup \N$, a partition $p\in \Pscr(k,l)$ is said to be of even size if $k+l$ is an even number, i.e., if $p$ has evenly many points.

It is sometimes said that the category of partitions of even size is the even part of the category $\Pscr$ of all partitions.

Canonical generator

The category of all partitions of even size is the subcategory of $\Pscr$ generated by the set of partitions $\{\crosspart,\fourpart,\singleton \otimes\singleton\}$.

Associated easy quantum group

Via Tannaka-Krein duality for compact quantum groups, the category of all partitions of even size corresponds to the family $(S'_N)_{N\in \N}$ of modified symmetric groups.

References


[BanSp09] Banica, Teodor and Speicher, Roland, 2009. Liberation of orthogonal Lie groups. Advances in Mathematics, 222, pp.1461–150.
category_of_partitions_of_even_size.txt · Last modified: 2021/11/23 11:56 (external edit)