Geodesic
Doubly stochastic matrices and the Bruhat order
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 681-700.

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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 $\Omega _n$ of doubly stochastic matrices (convex hull of $n\times n$ permutation matrices). An alternative description of this partial order is given. We define a class of special faces of $\Omega _n$ induced by permutation matrices, which we call Bruhat faces. Several examples of Bruhat faces are given and several results are presented.
DOI : 10.1007/s10587-016-0286-6
Classification : 05B20, 06A07, 15B51
Keywords: Bruhat order; doubly stochastic matrix; face 
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Brualdi, Richard A.; Dahl, Geir; Fritscher, Eliseu. Doubly stochastic matrices and the Bruhat order. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 681-700. doi : 10.1007/s10587-016-0286-6. http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0286-6/

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