non-group-theoretical_hyperoctahedral_categories_of_partitions
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non-group-theoretical_hyperoctahedral_categories_of_partitions [2020/01/27 07:32] amang [Definition] |
non-group-theoretical_hyperoctahedral_categories_of_partitions [2021/11/23 11:56] (current) |
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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//, meaning $|B|\in 2\N$ for every block $B$ of $p$ (see [[category of partitions with blocks of even size]]) . | ||
| * $p$ 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 [[category of partitions of even size|even size]], has [[category of partitions with blocks of even size|blocks of even size]] and has [[category of partitions with blocks of even size and even distances between legs|even distances between legs]]. Moreover, if $l\leq 1$, then it is also [[category of all non-crossing partitions|non-crossing]]. | + | 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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