Geodesic
Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010).

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We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth. 
@article{DMTCS_2010_special_259_a30,
     author = {Oh, Suho and Yoo, Hwanchul},
     title = {Bruhat order, rationally smooth {Schubert} varieties, and hyperplane arrangements},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)},
     year = {2010},
     doi = {10.46298/dmtcs.2835},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2835/}
}
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Oh, Suho; Yoo, Hwanchul. Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010). doi : 10.46298/dmtcs.2835. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2835/

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