It is well known that if a finite graded lattice of rank n is supersolvable, then it has an EL-labeling where the labels along any maximal chain form a permutation. We call such a labeling an Sn EL-labeling and we show that a finite graded lattice of rank n is supersolvable if and only if it has such a labeling. We next consider finite graded posets of rank n with 0̂ and 1̂ that have an Sn EL-lab...

Let W be a Coxeter group acting as a matrix group by way of the dual of the geometric representation. Let L be the lattice of intersections of all reflecting hyperplanes associated with the reflections in this representation. We show that L is isomorphic to the lattice consisting of all parabolic subgroups of W . We use this correspondence to find all W for which L is supersolvable. In particul...

We show that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a congruence normal lattice. Specifically, the poset of regions of a supersolvable arrangement of rank k is obtained via a sequence of doublings from the poset of regions of a supersolvable arrangement of rank k − 1. An explicit description of the doublings leads to a proof t...

Some posets of binary leaf-labeled trees are shown to be supersolvable lattices and explicit EL-labelings are given. Their characteristic polynomials are computed, recovering their known factorization in a different way.

Let G be a simple graph on the vertex set {v1, . . . , vn} with edge set E. Let K be a field. The graphical arrangement AG in K n is the arrangement xi − xj = 0, vivj ∈ E. An arrangement A is supersolvable if the intersection lattice L(c(A)) of the cone c(A) contains a maximal chain of modular elements. The second author has shown that a graphical arrangement AG is supersolvable if and only if ...

We introduce a poset structure on the reduced galleries in a supersolvable arrangement of hyperplanes. In particular, for Coxeter groups of type A or B, we construct a poset of reduced words for the longest element whose Hasse diagram is the graph of reduced words. Using Rambau’s Suspension Lemma, we show that these posets are homotopy equivalent to spheres. We furthermore conjecture that its i...