modified_bistochastic_group
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modified_bistochastic_group [2020/02/13 06:23] amang created |
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| ====== Modified bistochastic group ====== | ====== Modified bistochastic group ====== | ||
| - | A **modified bistochastic group** is any member of a certain sequence $(B_N')_{N\in \N}$ of [[classical matrix groups]]. | + | A **modified bistochastic group** is any member of a certain sequence $(B_N')_{N\in \N}$ of [[classical orthogonal matrix groups]]. |
| ===== Definition ===== | ===== Definition ===== | ||
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| For every $N\in \N$ the **modified bistochastic group** for dimension $N$ is the subgroup of the [[wp>general linear group]] $\mathrm{GL}(N,\C)$ given by all bistochastic [[orthogonal group|orthogonal]] $N\times N$-matrices multiplied by a factor of $1$ or $-1$, i.e., the set | For every $N\in \N$ the **modified bistochastic group** for dimension $N$ is the subgroup of the [[wp>general linear group]] $\mathrm{GL}(N,\C)$ given by all bistochastic [[orthogonal group|orthogonal]] $N\times N$-matrices multiplied by a factor of $1$ or $-1$, i.e., the set | ||
| $$B_N'\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I,\,\exists r\in\{-1,1\}: \forall_{i,j=1}^N: {\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=r\},$$ | $$B_N'\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I,\,\exists r\in\{-1,1\}: \forall_{i,j=1}^N: {\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=r\},$$ | ||
| - | 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 ===== | ===== Basic properties ===== | ||
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| + | The modified bistochastic 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 of even size with small blocks]] that induces the corepresentation categories of $(B_N')_{N\in \N}$. Its canonical generating set of partitions is $\{\crosspart,\singleton\otimes\singleton\}$. | ||
| ===== Representation theory ===== | ===== Representation theory ===== | ||
modified_bistochastic_group.1581575026.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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