Abstract
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.
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