The absolute order on the hyperoctahedral group

TL;DR

This paper studies the structure of the absolute order on the hyperoctahedral group, proving homotopy Cohen-Macaulayness, shellability, and characterizing lattice intervals, with explicit computations of invariants.

Contribution

It establishes new topological and combinatorial properties of the absolute order on $B_n$, including homotopy Cohen-Macaulayness and lattice characterizations.

Findings

Order ideal generated by Coxeter elements is homotopy Cohen-Macaulay

Every closed interval in the absolute order on $B_n$ is shellable

Characterization of lattice intervals and computation of invariants

Abstract

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.