A bistochastic group is any member of a sequence of classical orthogonal matrix groups.
For given any scalar -matrix is called
For every the bistochastic group for dimension is the subgroup of the general linear group given by all bistochastic orthogonal -matrices, i.e., the set
where, if , then is the complex conjugate of and the transpose and where is the identity -matrix.
Note that the elements of 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 -matrices forms a compact Hausdorff semigroup with respect to the topology inherited from – 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 [MonPle73].
The bistochastic groups 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 all partitions with small blocks that induces the corepresentation categories of . Its canonical generating set of partitions is .