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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... Theorem . Basis Extension The Map from D ( A ) to D ( A " ) The Addition - Deletion Theorem Inductively Free Arrangements . Supersolvable Arrangements 62 63 65 67 70 70 72 74 77 78 79 80 82 85 86 86 88 89 92 92 93 155 95 97 99 100 100 ...
... Theorem . Basis Extension The Map from D ( A ) to D ( A " ) The Addition - Deletion Theorem Inductively Free Arrangements . Supersolvable Arrangements 62 63 65 67 70 70 72 74 77 78 79 80 82 85 86 86 88 89 92 92 93 155 95 97 99 100 100 ...
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... Theorem 5. Topology 5.1 The Complement M ( A ) K ( π , 1 ) -Arrangements Free Arrangements Generic Arrangements · Deformation . Arnold's Conjectures 5.2 The Homotopy Type of M ( A ) Real Arrangements The Homotopy Type Complexified Real ...
... Theorem 5. Topology 5.1 The Complement M ( A ) K ( π , 1 ) -Arrangements Free Arrangements Generic Arrangements · Deformation . Arnold's Conjectures 5.2 The Homotopy Type of M ( A ) Real Arrangements The Homotopy Type Complexified Real ...
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... theorem of Brylawski [ 43 ] about the Poincaré polynomial under deletion and restriction : ( 2 ) π ( А , t ) = π ( A ' , t ) + tñ ( A ′′ , t ) . In Section 2.3 we also prove a theorem of Stanley [ 218 ] , which asserts that if L ( A ) ...
... theorem of Brylawski [ 43 ] about the Poincaré polynomial under deletion and restriction : ( 2 ) π ( А , t ) = π ( A ' , t ) + tñ ( A ′′ , t ) . In Section 2.3 we also prove a theorem of Stanley [ 218 ] , which asserts that if L ( A ) ...
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... theorems in Section 3.3 . If cA is the cone over A , then there exists an element ao Є A ( CA ) so that there is an ... Theorem , which asserts that if ( A , A ' , A " ) is a triple , then any two of the following statements imply the ...
... theorems in Section 3.3 . If cA is the cone over A , then there exists an element ao Є A ( CA ) so that there is an ... Theorem , which asserts that if ( A , A ' , A " ) is a triple , then any two of the following statements imply the ...
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... Theorem , which asserts that if A is a free l - arrangement with exp A = { b1 , ... , b1 } , then π ( A , t ) = ( 1 + b1t ) · · · ( 1 + bet ) . ... Thus the exponents of a free arrangement are determined by combinatorial data . In the ...
... Theorem , which asserts that if A is a free l - arrangement with exp A = { b1 , ... , b1 } , then π ( A , t ) = ( 1 + b1t ) · · · ( 1 + bet ) . ... Thus the exponents of a free arrangement are determined by combinatorial data . In the ...
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