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...

Let B be a subset of the set of all isomorphism classes of finite groups. We consider the number Fg(x) of positive integers n < x such that all groups of order n lie in B. When ir consists of the isomorphism classes of all finite groups of any of the following types, we obtain an asymptotic formula for Fn(x): cyclic groups, abelian groups, nilpotent groups, supersolvable groups, and solvable gr...

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 ...

The following paper examines properties of the weak order on the groups Sr+1 and Br+1 as motivated by the search for a combinatorial proof of known h-vector inequalities. Several general results are developed with respect to descent set domination, and an explanation of the computer programs used to generate a number of these results is presented. While this paper primarily focuses on hand flag...

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...

A positive integer n is called a CLT number if every group of order n satisfies the converse of Lagrange’s Theorem. In this note, we find all CLT and supersolvable numbers up to 1000. We also formulate some questions about the distribution of these numbers. RÉSUMÉ: Un nombre entier naturel n est appelé nombre CLT si tout groupe d’ordre n satisfait la réciproque du Théorme de Lagrange. Nous déte...