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modified_bistochastic_group

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Modified bistochastic group

A modified bistochastic group is any member of a certain sequence $(B_N')_{N\in \N}$ of classical orthogonal matrix groups.

Definition

As a matrix group

For given $N\in \N$ any scalar $N\times N$-matrix $u=(u_{i,j})_{i,j=1}^N\in \C^{N\times N}$ is called

  • right stochastic if $\sum_{\ell=1}^N u_{i,\ell}=1$ for all $i\in \{1,\ldots,N\}$, i.e., if each row of $u$ sums up to $1$,
  • left stochastic if $\sum_{k=1}^N u_{k,j}=1$ for all $j\in \{1,\ldots,N\}$, i.e., if each column of $u$ sums up to $1$,
  • bistochastic or doubly stochastic if $u$ is both right and left stochastic.

For every $N\in \N$ the modified bistochastic group for dimension $N$ is the subgroup of the general linear group $\mathrm{GL}(N,\C)$ given by all bistochastic orthogonal $N\times N$-matrices multiplied by a factor of $1$ or $-1$, i.e., the set

$$B_N'\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I,\,\exists r\in\{-1,1\}: \forall_{i,j=1}^N: {\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=r\},$$

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.

As a direct product

The modified bistochastic group for dimension $N$, where $N\in \N$, can also be defined as the direct product of groups $B_N'\colon\hspace{-0.66em}=\Z_2\times B_N$ of the cyclic group $\Z_2\equiv \Z/2\Z$ of order $2$ and the bistochastic group bistochastic group for dimension $N$.

Basic properties

The modified bistochastic groups $(B_N')_{N\in \N}$ are an 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

Cohomology

References

modified_bistochastic_group.txt · Last modified: 2021/11/23 11:56 (external edit)

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