A certain Bruhat order on doubly substochastic matrices.

Denote by $ \omega _n $ ω n the convex polytope of $ n\times n $ n × n doubly substochastic matrices and by $ \Omega _n $ Ω n the convex polytope of $ n\times n $ n × n doubly stochastic matrices. We generalize the Bruhat order for permutation matrices and doubly stochastic matrices to the subpermutation matrices and doubly substochastic matrices, respectively. Moreover we study the Bruhat shadow of a subpermutation matrix and the Bruhat faces of $ \omega _n $ ω n . The relations between the Bruhat faces in $ \omega _n $ ω n and those in $ \Omega _n $ Ω n are also given. [ABSTRACT FROM AUTHOR]