category_of_partitions_with_blocks_of_even_size_and_even_distances_between_legs
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category_of_partitions_with_blocks_of_even_size_and_even_distances_between_legs [2020/01/22 09:28] amang created |
category_of_partitions_with_blocks_of_even_size_and_even_distances_between_legs [2021/11/23 11:56] (current) |
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| - | ====== Category of partitions with blocks of even size and even distances between legs ====== | + | ====== Category of partitions with blocks of even size and parity-balanced legs ====== |
| - | The **category of partitions with blocks of even size and even distances between legs** is a [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[half-liberated hyperoctahedral quantum group|half-liberated hyperoctahedral quantum groups]]. | + | The **category of partitions with blocks of even size and parity-balanced legs** is a [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[half-liberated hyperoctahedral quantum group|half-liberated hyperoctahedral quantum groups]]. |
| ===== Definition ===== | ===== Definition ===== | ||
| - | By the **category of partitions with blocks of even size and even distances between legs** one denotes the subcategory of the [[category of all partitions]] $\Pscr$ whose morphism class is the //set of partitions with blocks of even size and even distances between legs//. It was introduced by Banica, Curran and Speicher in [(:ref:BanCuSp10)]. | + | By the **category of partitions with blocks of even size and parity-balanced legs** one denotes the subcategory of the [[category of all partitions]] $\Pscr$ whose morphism class is the //set of partitions with blocks of even size and even distances between legs//. It was introduced by Banica, Curran and Speicher in [(:ref:BanCuSp10)]. |
| A partition $p\in \Pscr$ belongs to this set if the following conditions are met: | A partition $p\in \Pscr$ belongs to this set if the following conditions are met: | ||
| * $p$ has **blocks of even size**, i.e., every block of $p$ has an even number of legs. | * $p$ has **blocks of even size**, i.e., every block of $p$ has an even number of legs. | ||
| - | * $p$ has **even distances between legs**. This property has been expressed in three different but equivalent ways: | + | * $p$ has **parity-balanced legs**, i.e., for any block of $p$ when counting from an arbitrary point $i$ of the partition, the number of legs of $B$ at even distances from $i$ is equal to the number of legs of $B$ at odd distances from $i$. |
| - | * For any given block $B$ of $p$ only evenly many blocks $B'$ of $p$ with $B\neq B'$ exist which cross $B$, i.e., such that one can find $i,j\in B$ and $i',j'\in B'$ with $i\prec i'\prec j$ and $i'\prec j\prec j'$ (where $\cdot\!\prec\!\cdot\!\prec\!\cdot$ is the cyclic order of $p$). | + | * The name **set of partitions with blocks of even size and parity-balanced legs** is to be taken literally. |
| - | * For any block $B$ of $p$ and any two legs $i,j\in B$ there is an even number of points located between $i$ and $j$, i.e. in the interval $]i,j[_p$ given by the set $\{ k\,\vert\, i\prec k\prec j\}$. | + | |
| - | * If one labels the points of $p$ in alternating fashion with one of two symbols $\oplus$ and $\ominus$ along the cyclic order of $p$, then blocks of $p$ may only join points with unequal labels. | + | |
| - | * The name **set of partitions with blocks of even size and even distances between legs** is to be taken literally. | + | |
| A partition with blocks of even size is in particular of even size itself. | A partition with blocks of even size is in particular of even size itself. | ||
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| ===== Canonical Generator ===== | ===== Canonical Generator ===== | ||
| - | The category of partitions with blocks of even size and even distances between legs is the subcategory of $\Pscr$ generated by the set $\{\Pabcabc,\fourpart\}$ of partitions. | + | The category of partitions with blocks of even size and parity-balanced legs is the subcategory of $\Pscr$ generated by the set $\{\Pabcabc,\fourpart\}$ of partitions. |
| ===== Associated easy quantum group ===== | ===== Associated easy quantum group ===== | ||
| - | Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the category of partitions with blocks of even size and even distances between legs corresponds to the family $(H^{\ast}_N)_{N\in \N}$ of [[half-liberated hyperoctahedral quantum group|half-liberated hyperoctahedral quantum groups]]. | + | Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the category of partitions with blocks of even size and parity-balanced legs corresponds to the family $(H^{\ast}_N)_{N\in \N}$ of [[half-liberated hyperoctahedral quantum group|half-liberated hyperoctahedral quantum groups]]. |
| ===== References ===== | ===== References ===== | ||
category_of_partitions_with_blocks_of_even_size_and_even_distances_between_legs.1579685313.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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