Arrangements of Hyperplanes
An arrangement of hyperplanes is a finite collection of codimension one affine subspaces in a finite dimensional vector space. Arrangements have emerged independently as important objects in various fields of mathematics such as combinatorics, braids, configuration spaces, representation theory, reflection groups, singularity theory, and in computer science and physics. This book is the first comprehensive study of the subject. It treats arrangements with methods from combinatorics, algebra, algebraic geometry, topology, and group actions. It emphasizes general techniques which illuminate the connections among the different aspects of the subject. Its main purpose is to lay the foundations of the theory. Consequently, it is essentially self-contained and proofs are provided. Nevertheless, there are several new results here. In particular, many theorems that were previously known only for central arrangements are proved here for the first time in completegenerality. The text provides the advanced graduate student entry into a vital and active area of research. The working mathematician will findthe book useful as a source of basic results of the theory, open problems, and a comprehensive bibliography of the subject.
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affine algebra arrangement assertion associated assume basis braid called central arrangement choose circuit cohomology complement complex compute consider consists construction contains Corollary corresponding Coxeter group defined Definition denote dependent Derg derivations determined discriminant elements equal equivalent exact Example exists face fact factorization fiber field Figure finite fixed 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 subspace supersolvable Suppose surjective Table Theorem theory topological unique vector vertex vertices write