Geodesic
The absolute order on the hyperoctahedral group
Journal of Algebraic Combinatorics, Tome 34 (2011) no. 2, pp. 183-211.

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Summary: The absolute order on the hyperoctahedral group $B _{ n }$ is investigated. Using a poset fiber theorem, it is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen-Macaulay. This method results in a new proof of Cohen-Macaulayness of the absolute order on the symmetric group. Moreover, it is shown that every closed interval in the absolute order on $B _{ n }$ is shellable and an example of a non-Cohen-Macaulay interval in the absolute order on $D _{4}$ is given. Finally, the closed intervals in the absolute order on $B _{ n }$ and $D _{ n }$ which are lattices are characterized and some of their important enumerative invariants are computed.
Keywords: keywords hyperoctahedral group, absolute order, Cohen-Macaulay poset 
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     author = {Kallipoliti, Myrto},
     title = {The absolute order on the hyperoctahedral group},
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Kallipoliti, Myrto. The absolute order on the hyperoctahedral group. Journal of Algebraic Combinatorics, Tome 34 (2011) no. 2, pp. 183-211. http://geodesic.mathdoc.fr/item/JAC_2011__34_2_a5/