Geodesic
Shortest path poset of finite Coxeter groups
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009).

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We define a poset using the shortest paths in the Bruhat graph of a finite Coxeter group $W$ from the identity to the longest word in $W, w_0$. We show that this poset is the union of Boolean posets of rank absolute length of $w_0$; that is, any shortest path labeled by reflections $t_1,\ldots,t_m$ is fully commutative. This allows us to give a combinatorial interpretation to the lowest-degree terms in the complete $\textbf{cd}$-index of $W$. 
@article{DMTCS_2009_special_256_a43,
     author = {Blanco, Sa\'ul A.},
     title = {Shortest path poset of finite {Coxeter} groups},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)},
     year = {2009},
     doi = {10.46298/dmtcs.2721},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2721/}
}
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%0 Journal Article
%A Blanco, Saúl A.
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%J Discrete mathematics & theoretical computer science
%D 2009
%V DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
%I mathdoc
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%R 10.46298/dmtcs.2721
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%F DMTCS_2009_special_256_a43
Blanco, Saúl A. Shortest path poset of finite Coxeter groups. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009). doi : 10.46298/dmtcs.2721. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2721/

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