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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 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 $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ 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.

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 $N$ .

Table of Contents

Modified bistochastic group

Definition

As a matrix group

As a direct product

Basic properties

Representation theory

Cohomology

Related quantum groups

References

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Table of Contents

Modified bistochastic group

Definition

As a matrix group

As a direct product

Basic properties

Representation theory

Cohomology

Related quantum groups

References

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