The absolute order on the hyperoctahedral group

Details The absolute order on the hyperoctahedral group The absolute order on the hyperoctahedral group

2010-02-02

arxiv.org/abs/1002.0440v2

Mathematics

Combinatorics

26 pages, 6 figures. Theorem 1.3 of the previous version of this paper is omitted due to a gap in the proof.

Scientific paper

The absolute order on the hyperoctahedral group $B_n$ is investigated. It is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen-Macaulay and the M\"obius number of this ideal is computed. 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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