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Bruhat order; doubly stochastic matrix; face
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.
[1] Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics 231 Springer, New York (2005). MR 2133266 | Zbl 1110.05001
[2] Björner, A., Brenti, F.: An improved tableau criterion for Bruhat order. Electron. J. Comb. 3 Research paper R22, 5 pages (1996), printed version J. Comb. 3 311-315 (1996). MR 1399399 | Zbl 0884.05096
[3] Brualdi, R. A.: Combinatorial Matrix Classes. Encyclopedia of Mathematics and Its Applications 108 Cambridge University Press, Cambridge (2006). MR 2266203 | Zbl 1106.05001
[4] Brualdi, R. A., Dahl, G.: The Bruhat shadow of a permutation matrix. Mathematical Papers in Honour of Eduardo Marques de Sá Textos de Matemática. Série B 39 Universidade de Coimbra, Coimbra (2006), 25-38. MR 2291012 | Zbl 1178.05023
[5] Brualdi, R. A., Deaett, L.: More on the Bruhat order for {$(0,1)$}-matrices. Linear Algebra Appl. 421 (2007), 219-232. MR 2294337 | Zbl 1161.05018
[6] Brualdi, R. A., Hwang, S.-G.: A Bruhat order for the class of {$(0,1)$}-matrices with row sum vector {$R$} and column sum vector {$S$}. Electron. J. Linear Algebra (electronic only) 12 (2004/2005), 6-16. MR 2139456
[7] Magyar, P.: Bruhat order for two flags and a line. J. Algebr. Comb. 21 (2005), 71-101. DOI 10.1007/s10801-005-6281-x | MR 2130795 | Zbl 1076.14069
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