Arrangements of Hyperplanes

affine algebra arrangement assertion associated assume basis braid called central arrangement choose circuit complement complex compute consider consists construction contains Corollary corresponding Coxeter group defined Definition denote dependent derivations Ders determined discriminant elements equal equivalent exact Example exists face fact factorization fiber field Figure finite follows follows from Lemma formula free arrangement function given gives graded graph hence homogeneous hyperplanes ideal implies independent induction integers irreducible isomorphism K-module lattice Lemma Let G linear linearly lines Math matrix maximal modular natural Note obtained Orbits partition plane Poincaré polynomial points polynomial poset positive projection Proof properties Proposition prove rank real arrangement Recall reflection arrangement reflection group restriction result satisfies sequence space supersolvable Suppose surjective Table Theorem theory unique vector vertices write