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7页
... integers called its exponents , exp A = { b1 , ... , be ) . These integers are unique up to order , but they are not necessarily distinct . Terao [ 226 ] proved several results concerning a triple ( A , A ' , A " ) . These results may ...
... integers called its exponents , exp A = { b1 , ... , be ) . These integers are unique up to order , but they are not necessarily distinct . Terao [ 226 ] proved several results concerning a triple ( A , A ' , A " ) . These results may ...
第10页
... integers Z , rational numbers Q , real numbers IR , complex numbers C , the field of q elements IF , and arbitrary fields IK , L. Definition 1.1 Let K be a field and let V be a vector space of dimension l . A hyperplane H in VK is an ...
... integers Z , rational numbers Q , real numbers IR , complex numbers C , the field of q elements IF , and arbitrary fields IK , L. Definition 1.1 Let K be a field and let V be a vector space of dimension l . A hyperplane H in VK is an ...
第17页
... integers . We close the chapter with a section on the connections between arrangements and graph theory . This section includes a discussion of the chromatic polynomial , a precursor of the characteristic and Poincaré polynomials ...
... integers . We close the chapter with a section on the connections between arrangements and graph theory . This section includes a discussion of the chromatic polynomial , a precursor of the characteristic and Poincaré polynomials ...
第18页
... integers , called its exponents , exp A = { b1 , ... , be } . These integers are unique up to order , but they are not necessarily distinct . In Section 4.3 we prove the Addition- Deletion Theorem , which asserts that if ( A , A ' , A ...
... integers , called its exponents , exp A = { b1 , ... , be } . These integers are unique up to order , but they are not necessarily distinct . In Section 4.3 we prove the Addition- Deletion Theorem , which asserts that if ( A , A ' , A ...
第19页
... 5.3 . It generalizes Randell's presentation of the fundamental group of the complexifi- cation of a real arrangement . In Section 5.4 we consider the cohomology groups of M ( A ) with integer coefficients . We 1.3 Outline 19 Topology.
... 5.3 . It generalizes Randell's presentation of the fundamental group of the complexifi- cation of a real arrangement . In Section 5.4 we consider the cohomology groups of M ( A ) with integer coefficients . We 1.3 Outline 19 Topology.
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