non-group-theoretical_hyperoctahedral_categories_of_partitions
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| ====== Non-group-theoretical hyperoctahedral categories of partitions ====== | ====== Non-group-theoretical hyperoctahedral categories of partitions ====== | ||
| - | The **non-group-group-theoretical hyperoctahedral categories** are a one-parameter family of [[category_of_partitions|Banica-Speicher categories of partitions]], indexed by the natural numbers with infinity, introduced by Raum and Weber in [(:ref:RaWe16)]. | + | The **non-group-theoretical hyperoctahedral categories** are a one-parameter family of [[category_of_partitions|Banica-Speicher categories of partitions]], indexed by the natural numbers with infinity, introduced by Raum and Weber in [(:ref:RaWe16)]. |
| ===== Definition ===== | ===== Definition ===== | ||
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| 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: | 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 //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 **even distances between legs**. This property has been expressed in three different but equivalent ways: | + | |
| - | * 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 $p$). | + | |
| - | * 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. | + | |
| * $p$ 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. | * $p$ 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 | * $p$ 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 | ||
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| * where $Y_1$, $Y_2$ and $Y_3$ contain none of the letters $a_1,\ldots,a_k$. | * 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//. Moreover, if $l\leq 1$, then it is also //non-crossing// (see [[category of all non-crossing partitions]].) | + | Any such partition is necessarily of [[category of partitions of even size|even size]] and has [[category of partitions with blocks of even size and even distances between legs|parity-balanced legs]]. Moreover, if $l\leq 1$, then it is also [[category of all non-crossing partitions|non-crossing]]. |
| ===== Canonical Generator ===== | ===== Canonical Generator ===== | ||
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| ===== Associated easy quantum group ===== | ===== Associated easy quantum group ===== | ||
| - | Via [[tannaka_krein_duality|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 group|non-group-theoretical hyperoctahedral easy quantum groups]]. | + | Via [[tannaka_krein_duality|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_orthogonal_quantum_groups|non-group-theoretical hyperoctahedral easy quantum groups]]. |
| ===== References ===== | ===== References ===== | ||
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