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

===== Cohomology =====

===== Related quantum groups =====

===== References =====