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--- hyperoctahedral_group [2020/02/13 06:35]
amang
+++ hyperoctahedral_group [2021/11/23 11:56] (current)
@@ Line 8: / Line 8: @@
 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
 $$H_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I, \,\forall_{i,j=1}^N:  u_{i,j}\in\Z\},$$
-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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 ===== Basic properties =====
+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 $\{\crosspart,\fourpart\}$.
 ===== Representation theory =====