Search

A family of hyperplane arrangements with one parameter contradicts K Saito's hypothesis that K(pi, 1) comprises the complexified complement of free arrangements. This conjecture comes from research into symmetric octagons and tilings. The generalization of supersolvable and Coxeter arrangements is a free arrangement. The braid arrangement A is K(pi, 1) where pi represents the pure braid group. Two theorems are presented, with sketched proofs.

We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders. [ABSTRACT FROM AUTHOR]

For finite reflection groups of types A and B, we determine the diameter of the graph whose vertices are reduced words for the longest element and whose edges are braid relations. This is deduced from a more general theorem that applies to supersolvable hyperplane arrangements. [ABSTRACT FROM AUTHOR]