Doubly stochastic matrices and the Bruhat order

On a conjecture concerning the Bruhat order

“…Since it is possible to construct the matrix Σ(X), for all real matrix X, the Bruhat order was extended to other classes of matrices in a natural way, [4,8]: let A, C be m-by-n real matrices with the same row sum vectors and the same column sum vectors then A precedes C in the Bruhat order, written A B C, provided that Σ(A) ≥ Σ(C) (by the entrywise order).…”

“…To prove this fact, we generalize the notion of an interchange from a (0, 1)-matrix to a real matrix. Note that this generalization to a doubly stochastic matrix can be seen in [4].…”

The Bruhat order on classes of isotopic Latin squares

“…The Bruhat order on (0, 1)-matrices is receiving the attention of many researchers, [4,6,10,11,12,13,15,16,17,18,19]. In the recent years several authors have taken Brualdi and Hwang's ideas, and extended the Bruhat order to other classes of matrices than (0, 1)-matrices: the Bruhat order has been studied on the class of tournament matrices with a given score vector, [8], on the class of alternating sign matrices, [9], and on the class of doubly stochastic matrices, [7]. In [14], the authors, followed this new branch of investigation, studied the Bruhat order on the class of Latin squares of order n. This partial order relation is defined similarly as it is defined on other classes of matrices, that is, by using the matrices Σ(.)…”

Classes of (0,1)-matrices Where the Bruhat Order and the Secondary Bruhat Order Coincide

“…These two partial orders have been intensively investigated in the recent years. The research focuses on several topics: minimal elements, [3] and [4], chains and antichains, [10], [9], [15], and [16], restrictions of the Bruhat order to some other classes of (0, 1)-matrices, [8], and [11], or extensions of the Bruhat order to other classes of matrices than (0, 1)-matrices, [5], [6], [7], and [13].…”