Doubly stochastic matrices and the Bruhat order

Abstract

The Bruhat order is defined in terms of an interchange operation on the set of permutation matrices of order n which corresponds to the transposition of a pair of elements in a permutation. We introduce an extension of this partial order, which we call the stochastic Bruhat order, for the larger class Ω n of doubly stochastic matrices (convex hull of n×n permutation matrices). An alternative description of this partial order is given. We define a class of special faces of Ω n induced by permutation matrices, which we call Bruhat faces. Several examples of Bruhat faces are given and several results are presented.