Arrangements of HyperplanesSpringer Science & Business Media, 2013年3月9日 - 325页 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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第vii页
... vector space . In this book we study arrange- ments with methods from combinatorics , algebra , algebraic geometry , topology , and group actions . These first sentences illustrate the two aspects of our subject that attract us most ...
... vector space . In this book we study arrange- ments with methods from combinatorics , algebra , algebraic geometry , topology , and group actions . These first sentences illustrate the two aspects of our subject that attract us most ...
第3页
... vector space VR of dimension l . Let V be the complexification of VR . Then W acts as a reflection group in V. Let Mw CV be the complement of the reflecting hyperplanes of W. He proved that the analog of ( 1 ) involves the exponents m1 ...
... vector space VR of dimension l . Let V be the complexification of VR . Then W acts as a reflection group in V. Let Mw CV be the complement of the reflecting hyperplanes of W. He proved that the analog of ( 1 ) involves the exponents m1 ...
第7页
... vector space V of dimension l , then it has exponents mi , ... me . However , if MG CV is the complement of the reflecting hyperplanes of G , then formula ( 3 ) does not hold for MG . Orlik and Solomon [ 172 ] defined coexponents n1 ...
... vector space V of dimension l , then it has exponents mi , ... me . However , if MG CV is the complement of the reflecting hyperplanes of G , then formula ( 3 ) does not hold for MG . Orlik and Solomon [ 172 ] defined coexponents n1 ...
第10页
... vector space of dimension l . A hyperplane H in VK is an affine subspace of dimension ( l - 1 ) . A hyper- plane arrangement AK = ( AK , VIK ) is a finite set of hyperplanes in VK . More generally , a subspace arrangement is a finite ...
... vector space of dimension l . A hyperplane H in VK is an affine subspace of dimension ( l - 1 ) . A hyper- plane arrangement AK = ( AK , VIK ) is a finite set of hyperplanes in VK . More generally , a subspace arrangement is a finite ...
第13页
... vector space over the finite field of q elements , Fq . Let A be the central arrangement in V consisting of all hyperplanes through the origin . Basic Constructions Definition 1.11 Let | A | denote the cardinality of A. In Example 1.8 ...
... vector space over the finite field of q elements , Fq . Let A be the central arrangement in V consisting of all hyperplanes through the origin . Basic Constructions Definition 1.11 Let | A | denote the cardinality of A. In Example 1.8 ...
目录
1 | |
4 | |
15 | |
22 | |
Examples | 30 |
Algebras | 59 |
The Injective Map AAx AA | 65 |
Supersolvable Arrangements | 80 |
115 | 204 |
Reflection Arrangements | 215 |
A Some Commutative Algebra 271 | 270 |
B Basic Derivations | 279 |
Orbit Types | 289 |
ThreeDimensional Restrictions | 301 |
164 | 310 |
Index | 315 |
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常见术语和短语
A₁ affine arrangement algebra A(A arrangement and let assume b₁ basic derivations basic invariants basis for D(A bijection braid arrangement broken circuit called central arrangement choose cohomology complement complex reflection groups complexification compute Corollary Coxeter group defining polynomial Definition deformation retraction degree denote exact sequence Example exterior product fiber finite follows from Lemma follows from Proposition follows from Theorem formula free arrangement free with exp fundamental group G-orbit graph H₁ H₂ homogeneous homotopy type hyperplane arrangement inductively free integers irreducible isomorphism ker(x l-arrangement lattice Lemma Let G linear linearly independent Math matrix maximal element Möbius function modular elements nonempty Note Orbits Poincaré polynomial poset Proof prove real arrangement Recall reflection arrangement restriction result Section set of basic Shephard groups simplicial subset subspace supersolvable Suppose surjective topological unitary reflection group vertex w₁ write