Topics in hyperplane arrangements
 Responsibility
 Marcelo Aguiar, Swapneel Mahajan.
 Publication
 Providence, Rhode Island : American Mathematical Society, [2017]
 Copyright notice
 ©2017
 Physical description
 xxiv, 611 pages : illustrations ; 27 cm.
 Series
 Mathematical surveys and monographs ; no. 226.
Access
Available online
Science Library (Li and Ma)
Stacks
Call number  Status 

QA3 .A4 V.226  Unknown 
More options
Creators/Contributors
 Author/Creator
 Aguiar, Marcelo, 1968 author.
 Contributor
 Mahajan, Swapneel Arvind, 1974 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 579595) and indexes.
 Contents

 Part I: Hyperplane arrangementsConesLunesCategory of lunesReflection arrangementsBraid arrangement and related examplesDescent and lune equationsDistance functions and Varchenko matrixPart II: Birkhoff algebra and Tits algebraLie and Zie elementsEulerian idempotentsDiagonalizability and characteristic elementsLoewy series and Peirce decompositionsDynkin idempotentsIncidence algebrasInvariant Birkhoff algebra and invariant Tits algebraAppendices: Regular cell complexesPosetsIncidence algebras of posetsAlgebras and modulesBandsReferences: BibliographyNotation indexSubject index.
 (source: Nielsen Book Data)9781470437114 20180115
 Publisher's Summary
 This monograph studies the interplay between various algebraic, geometric and combinatorial aspects of real hyperplane arrangements. It provides a careful, organized and unified treatment of several recent developments in the field, and brings forth many new ideas and results. It has two parts, each divided into eight chapters, and five appendices with background material. Part I gives a detailed discussion on faces, flats, chambers, cones, gallery intervals, lunes and other geometric notions associated with arrangements. The Tits monoid plays a central role. Another important object is the category of lunes which generalizes the classical associative operad. Also discussed are the descent and lune identities, distance functions on chambers, and the combinatorics of the braid arrangement and related examples. Part II studies the structure and representation theory of the Tits algebra of an arrangement. It gives a detailed analysis of idempotents and Peirce decompositions, and connects them to the classical theory of Eulerian idempotents. It introduces the space of Lie elements of an arrangement which generalizes the classical Lie operad. This space is the last nonzero power of the radical of the Tits algebra. It is also the socle of the left ideal of chambers and of the right ideal of Zie elements. Zie elements generalize the classical Lie idempotents. They include Dynkin elements associated to generic halfspaces which generalize the classical Dynkin idempotent. Another important object is the luneincidence algebra which marks the beginning of noncommutative Mobius theory. These ideas are also brought upon the study of the Solomon descent algebra. The monograph is written with clarity and in sufficient detail to make it accessible to graduate students. It can also serve as a useful reference to experts.
(source: Nielsen Book Data)9781470437114 20180115
Subjects
 Subject
 Incidence algebras.
 Algebraic spaces.
 Hyperspace.
 Geometry, Plane.
 Algebraic spaces.
 Geometry, Plane.
 Hyperspace.
 Incidence algebras.
 Combinatorics > Algebraic combinatorics > Combinatorial aspects of representation theory.
 Order, lattices, ordered algebraic structures > Ordered sets > Combinatorics of partially ordered sets.
 Order, lattices, ordered algebraic structures > Ordered sets > Algebraic aspects of posets.
 Order, lattices, ordered algebraic structures > Modular lattices, complemented lattices > Semimodular lattices, geometric lattices.
 Associative rings and algebras > Representation theory of rings and algebras > Representations of Artinian rings.
 Nonassociative rings and algebras > Lie algebras and Lie superalgebras > Identities, free Lie (super)algebras.
 Group theory and generalizations > Special aspects of infinite or finite groups > Reflection and Coxeter groups.
 Group theory and generalizations > Semigroups > General structure theory.
 Group theory and generalizations > Semigroups > Semigroup rings, multiplicative semigroups of rings.
 Convex and discrete geometry > Discrete geometry > Arrangements of points, flats, hyperplanes.
 Combinatorics > Enumerative combinatorics > Partitions of sets.
 Combinatorics > Enumerative combinatorics > $q$calculus and related topics.
 Combinatorics > Graph theory > Graphs and abstract algebra (groups, rings, fields, etc.)
 Combinatorics > Algebraic combinatorics > Symmetric functions and generalizations.
 Combinatorics > Algebraic combinatorics > Group actions on combinatorial structures.
 Combinatorics > Algebraic combinatorics > Combinatorial aspects of simplicial complexes.
 Category theory; homological algebra > Categories with structure > Operads.
 Category theory; homological algebra > Homological algebra > Chain complexes.
 Group theory and generalizations > Permutation groups > Symmetric groups.
 Combinatorics  Algebraic combinatorics  Combinatorial aspects of representation theory.
 Order, lattices, ordered algebraic structures  Ordered sets  Combinatorics of partially ordered sets.
 Order, lattices, ordered algebraic structures  Ordered sets  Algebraic aspects of posets.
 Order, lattices, ordered algebraic structures  Modular lattices, complemented lattices  Semimodular lattices, geometric lattices.
 Associative rings and algebras  Representation theory of rings and algebras  Representations of Artinian rings.
 Nonassociative rings and algebras  Lie algebras and Lie superalgebras  Identities, free Lie (super)algebras.
 Group theory and generalizations  Special aspects of infinite or finite groups  Reflection and Coxeter groups.
 Group theory and generalizations  Semigroups  General structure theory.
 Group theory and generalizations  Semigroups  Semigroup rings, multiplicative semigroups of rings.
 Convex and discrete geometry  Discrete geometry  Arrangements of points, flats, hyperplanes.
 Combinatorics  Enumerative combinatorics  Partitions of sets.
 Combinatorics  Enumerative combinatorics  $q$calculus and related topics.
 Combinatorics  Graph theory  Graphs and abstract algebra (groups, rings, fields, etc.)
 Combinatorics  Algebraic combinatorics  Symmetric functions and generalizations.
 Combinatorics  Algebraic combinatorics  Group actions on combinatorial structures.
 Combinatorics  Algebraic combinatorics  Combinatorial aspects of simplicial complexes.
 Category theory; homological algebra  Categories with structure  Operads.
 Category theory; homological algebra  Homological algebra  Chain complexes.
 Group theory and generalizations  Permutation groups  Symmetric groups.
 Algebraic spaces.
 Geometry, Plane.
 Hyperspace.
 Incidence algebras.
Bibliographic information
 Publication date
 2017
 Copyright date
 2017
 Series
 Mathematical surveys and monographs ; volume 226
 ISBN
 9781470437114 hardcover
 1470437112 hardcover