hyperoctahedral_group
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hyperoctahedral_group [2020/02/13 06:33] amang [Hyperoctahedral group] |
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| ====== Hyperoctahedral group ====== | ====== Hyperoctahedral group ====== | ||
| - | A **hyperoctahedral group** is any member of a sequence $(H_N)_{N\in \N}$ of [[classical matrix group|classical matrix groups]]. | + | A **hyperoctahedral group** is any member of a sequence $(H_N)_{N\in \N}$ of [[classical orthogonal matrix groups]]. |
| ===== Definition ===== | ===== Definition ===== | ||
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| For every $N\in \N$ the hyperoctahedral group $H_N$ for dimension $N$ can be defined as the subgroup of the [[wp>general linear group]] $\mathrm{GL}(N,\C)$ given by all orthogonal $N\times N$-matrices with integer entries, i.e., the set | For every $N\in \N$ the hyperoctahedral group $H_N$ for dimension $N$ can be defined as the subgroup of the [[wp>general linear group]] $\mathrm{GL}(N,\C)$ given by all orthogonal $N\times N$-matrices with integer entries, i.e., the set | ||
| $$H_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I, \,\forall_{i,j=1}^N: u_{i,j}\in\Z\},$$ | $$H_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I, \,\forall_{i,j=1}^N: u_{i,j}\in\Z\},$$ | ||
| - | where, if $u=(u_{i,j})_{i,j=1}^N$, then $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose and where $I_N$ is the identity $N\!\times \!N$-matrix. | + | where, if $u=(u_{i,j})_{i,j=1}^N$, then $\overline u=(\overline{u}_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose and where $I_N$ is the identity $N\!\times \!N$-matrix. |
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| For the hyperoctahedral group $H_N$ this means that $H_N=\Z_2\wr S_N$ is given by the group with underlying set $(\Z_2)^{\times N}\times S_N$ and group law $(((i_1,\ldots,i_N),\pi),((j_1,\ldots,j_N),\rho))\mapsto ((i_1+j_{\pi^{-1}(1)},\ldots,i_N+j_{\pi^{-1}(N)}),\pi\circ \rho)$. | For the hyperoctahedral group $H_N$ this means that $H_N=\Z_2\wr S_N$ is given by the group with underlying set $(\Z_2)^{\times N}\times S_N$ and group law $(((i_1,\ldots,i_N),\pi),((j_1,\ldots,j_N),\rho))\mapsto ((i_1+j_{\pi^{-1}(1)},\ldots,i_N+j_{\pi^{-1}(N)}),\pi\circ \rho)$. | ||
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| + | ===== Basic properties ===== | ||
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| + | The hyperoctahedral groups $(B_N)_{N\in \N}$ are an [[easy_quantum_group|easy]] family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a [[category of partitions]]. More precisely, it is the [[category of partitions with blocks of even size]] that induces the corepresentation categories of $(H_N)_{N\in \N}$. Its canonical generating set of partitions is $\{\crosspart,\fourpart\}$. | ||
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| + | ===== Representation theory ===== | ||
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| + | ===== Cohomology ===== | ||
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| + | ===== Related quantum groups ===== | ||
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| + | ===== References ===== | ||
hyperoctahedral_group.1581575627.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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