Geodesic
Shortest path poset of Bruhat intervals
Journal of Algebraic Combinatorics, Tome 38 (2013) no. 3, pp. 585-596.

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Summary: We define the shortest path poset $SP(u,v)$ of a Bruhat interval [u,v], by considering the shortest u-v paths in the Bruhat graph of a Coxeter group W, where u,v$\in W$. We consider the case of $SP(u,v)$ having a unique rising chain under a reflection order and show that in this case $SP(u,v)$ is a $Gorenstein^{\ast }$ poset. This allows us to derive the nonnegativity of certain coefficients of the complete cd-index. We furthermore show that the shortest path poset of an irreducible, finite Coxeter group exhibits a symmetric chain decomposition.
Keywords: shortest paths, Bruhat graph, Bruhat order, $\widetilde{R}$ -polynomials, complete cd-index 
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     author = {Blanco, Sa\'ul A.},
     title = {Shortest path poset of {Bruhat} intervals},
     journal = {Journal of Algebraic Combinatorics},
     pages = {585--596},
     publisher = {mathdoc},
     volume = {38},
     number = {3},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2013__38_3_a6/}
}
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Blanco, Saúl A. Shortest path poset of Bruhat intervals. Journal of Algebraic Combinatorics, Tome 38 (2013) no. 3, pp. 585-596. http://geodesic.mathdoc.fr/item/JAC_2013__38_3_a6/