====== Bistochastic group ======

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

===== Definition =====

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 **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, i.e., the set
$$B_N\colon\hspace{-0.6em}=\{ u\in \C^{N\times N}\,\vert\, u=\overline{u},\, uu^t=u^tu=I, \, \forall_{i,j=1}^N: {\textstyle\sum_{\ell=1}^N} u_{i,\ell}={\textstyle\sum_{k=1}^N} u_{k,j}=1\},$$
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.

Note that the elements of $B_N$ are __not__ required to have non-negative entries. Stochastic matrices with non-negative entries are known as **probability matrices**, **transition matrices**, **substitution matrices** or **Markov matrices**. The set of such $N\times N$-matrices forms a compact Hausdorff semigroup with respect to the topology inherited from $\C^{N\times N}$ -- but not a group. In fact, a bistochastic matrix with non-negative entries has a bistochastic inverse with non-negative entries if and only if it is a permutation matrix [(:ref:MonPle73)].

===== Basic properties =====

The 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 all partitions with small blocks]] that induces the corepresentation categories of $(B_N)_{N\in \N}$. Its canonical generating set of partitions is $\{\crosspart,\singleton\}$. 

===== Representation theory =====

===== Cohomology =====

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

===== References =====

[( :ref:MonPle73 >>
author:  Montague, J.S. and Plemmons, R.J.
title:   Doubly stochastic matrix equations
year:    1973
journal: Israel Journal of Mathematics
volume:  15
issue:   3
pages:   216-229
url:     https://doi.org/10.1007/BF02787568
)]

[( :ref:BanSp09 >>
author:  Banica, Teodor and Speicher, Roland
title:   Liberation of orthogonal Lie groups
year:    2009
journal: Advances in Mathematics
volume:  222
issue:   4
pages:   1461--150
url:     https://doi.org/10.1016/j.aim.2009.06.009
archivePrefix: arXiv
eprint   :0808.2628
)]