q-algebras and arrangements of hyperplanes

Inductively Factored Signed-graphic Arrangements of Hyperplanes

In 1994, Edelman and Reiner characterized free and supersolvable hyperplane arrangements in the restricted interval [An−1, Bn]. In this paper, we give a characterization of inductively factored arrangements in this interval, and show that the same characterization also describes factored arrangements in this interval. These results use the compact notation of signed graphs introduced by Zaslavsky.

Chambers of Arrangements of Hyperplanes and Arrow’s Impossibility Theorem

Let A be a nonempty real central arrangement of hyperplanes and Ch be the set of chambers of A. Each hyperplane H defines a half-space H and the other half-space H. Let B = {+,−}. For H ∈ A, define a map ǫ H : Ch → B by ǫ H (C) = + (if C ⊆ H) and ǫ H (C) = − (if C ⊆ H). Define ǫ H = −ǫ H . Let Ch = Ch×Ch× · · · ×Ch (m times). Then the maps ǫ H induce the maps ǫ H : Ch → B. We will study the adm...

Generalized Catalan Numbers, Weyl Groups and Arrangements of Hyperplanes

For an irreducible, crystallographic root system Φ in a Euclidean space V and a positive integer m, the arrangement of hyperplanes in V given by the affine equations (α, x)= k, for α∈Φ and k=0, 1, . . . ,m, is denoted here by AΦ . The characteristic polynomial of AΦ is related in the paper to that of the Coxeter arrangement AΦ (corresponding to m=0), and the number of regions into which the fun...

Arrangements of Hyperplanes and Vector Bundles on P

Monodromy of hypergeometric functions arising from arrangements of hyperplanes

Given an arrangement of hyperplanes in P, possibly with non-normal crossings, we give a vanishing lemma for the cohomology of the sheaf of q-forms with logarithmic poles along our arrangement. We give a basis for the ideal J of relations for the Orlik-Solomon’s algebra. Under certain genericity conditions it was shown by H. Esnault, V. Schechtman and E. Viehweg that the cohomology of a local sy...