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 half-spaces which generalize the classical Dynkin idempotent. Another important object is the lune-incidence algebra which marks the beginning of noncommutative Möbius 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
Author(s): Marcelo Aguiar; Swapneel Arvind Mahajan
Series: Mathematical Surveys and Monographs 226
Publisher: AMS
Year: 2017
Language: English
Commentary: decrypted from 0088257EBD0476FF593193C76E3B873F source file
Pages: 639
Cover
Title page
Contents
Preface
Introduction
Part I.
Chapter 1. Hyperplane arrangements
1.1. Faces
1.2. Arrangements of small rank
1.3. Flats
1.4. Tits monoid and Birkhoff monoid
1.5. Bi-faces and Janus monoid
1.6. Order-theoretic properties of faces and flats
1.7. Arrangements under and over a flat
1.8. Cartesian product of arrangements
1.9. Generic hyperplanes and adjoints of arrangements
1.10. Separating hyperplanes, minimal galleries and gate property
1.11. Combinatorially isomorphic arrangements
1.12. Partial order on pairs of faces
1.13. Characteristic polynomial and Zaslavsky formula
Notes
Chapter 2. Cones
2.1. Cones and convexity
2.2. Case and base maps
2.3. Topology of a cone
2.4. Cutting and separating hyperplanes and gated sets
2.5. Gallery intervals
2.6. Charts and dicharts
2.7. Poset of top-cones
2.8. Partial-flats
Notes
Chapter 3. Lunes
3.1. Lunes
3.2. Nested faces and lunes
3.3. Decomposition of a cone into lunes
3.4. Restriction and extension of cones
3.5. Top-star-lunes
3.6. Conjugate top-cones
3.7. Cartesian product of cones, gallery intervals and lunes
Notes
Chapter 4. Category of lunes
4.1. Poset of top-lunes
4.2. Two partial orders on lunes
4.3. Maps involving lunes
4.4. Category of lunes
4.5. Categories associated to faces and flats
4.6. Presentation of categories
4.7. Action of the Birkhoff monoid on lunes
4.8. Substitution product of chambers
Notes
Chapter 5. Reflection arrangements
5.1. Coxeter groups and reflection arrangements
5.2. Face-types, flat-types and lune-types
5.3. Length, ?-valued distance and weak order
5.4. Subgroups of Coxeter groups
5.5. Cycle-type function and characteristic polynomial
5.6. Coxeter-Tits monoid
5.7. Good reflection arrangements
Notes
Chapter 6. Braid arrangement and related examples
6.1. Coordinate arrangement
6.2. Rank-two arrangements
6.3. Braid arrangement. Compositions and partitions
6.4. Braid arrangement. Partial orders and graphs
6.5. Braid arrangement. Linear compositions, partitions and shuffles
6.6. Enumeration in the braid arrangement
6.7. Arrangement of type ?
6.8. Arrangement of type ?
6.9. Graphic arrangements
Notes
Chapter 7. Descent and lune equations
7.1. Descent equation
7.2. Lune equation
7.3. Witt identities
7.4. Descent-lune equation for flats
7.5. Descent and lune equations for partial-flats
7.6. Faces and flats for left Σ-sets
7.7. Descent equation for left Σ-sets
7.8. Lune equation for left Σ-sets
7.9. Lune equation for right Σ-sets
7.10. Descent-lune equation for Π-sets
7.11. Flat-based lattices
Notes
Chapter 8. Distance functions and Varchenko matrix
8.1. Weights on half-spaces
8.2. Sampling weights from a matrix
8.3. Distance functions
8.4. Varchenko matrix
8.5. Symmetric Varchenko matrix
8.6. Braid arrangement
8.7. Type ? arrangement
Notes
Part II.
Chapter 9. Birkhoff algebra and Tits algebra
9.1. Birkhoff algebra
9.2. Algebras of charts, dicharts and cones
9.3. Tits algebra
9.4. Left module of chambers
9.5. Modules over the Tits algebra
9.6. Filtration by flats of a right module
9.7. Primitive part and decomposable part
9.8. Over and under a flat. Cartesian product
9.9. Janus algebra and its one-parameter deformation
9.10. Coxeter-Tits algebra
Notes
Chapter 10. Lie and Zie elements
10.1. Lie elements
10.2. Lie in small ranks. Antisymmetry and Jacobi identity
10.3. Zie elements
10.4. Zie elements and primitive part of modules
10.5. Zie in small ranks
10.6. Substitution product of Lie
Notes
Chapter 11. Eulerian idempotents
11.1. Homogeneous sections of the support map
11.2. Eulerian idempotents
11.3. Eulerian families, complete systems and algebra sections
11.4. Q-bases of the Tits algebra
11.5. Families of Zie idempotents
11.6. Eulerian idempotents for good reflection arrangements
11.7. Extension problem and dimension of Lie
11.8. Rank-two arrangements
11.9. Rank-three arrangements
Notes
Chapter 12. Diagonalizability and characteristic elements
12.1. Stationary distribution
12.2. Diagonalizability and eigensections
12.3. Takeuchi element
12.4. Characteristic elements
12.5. Type ? Eulerian idempotents and Adams elements
12.6. Type ? Eulerian idempotents and Adams elements
Notes
Chapter 13. Loewy series and Peirce decompositions
13.1. Primitive series and decomposable series
13.2. Primitive series and socle series
13.3. Radical series and primitive series
13.4. Peirce decompositions, and primitive and decomposable series
13.5. Left Peirce decomposition of chambers. Lie over flats
13.6. Right Peirce decomposition of Zie. Lie under flats
13.7. Two-sided Peirce decomposition of faces. Lie over & under flats
13.8. Generation of Lie elements in rank one
13.9. Rigidity of the left module of chambers
13.10. Quiver of the Tits algebra
13.11. Applications of Peirce decompositions to Loewy series
Notes
Chapter 14. Dynkin idempotents
14.1. Dynkin elements
14.2. Dynkin basis for the space of Lie elements
14.3. Applications to affine hyperplane arrangements
14.4. Orientation space
14.5. Joyal-Klyachko-Stanley. Presentation of Lie
14.6. Björner and Lyndon bases
14.7. Coordinate arrangement
14.8. Rank-two arrangements
14.9. Classical (type ?) Lie elements
14.10. Type ? Lie elements
Notes
Chapter 15. Incidence algebras
15.1. Flat-incidence algebra
15.2. Lune-incidence algebra
15.3. Noncommutative zeta and Möbius functions
15.4. Noncommutative Möbius inversion. Group-likes and primitives
15.5. Characterizations of Eulerian families
15.6. Lie-incidence algebra
15.7. Additive and Weisner functions on lunes
15.8. Subalgebras of the lune-incidence algebra
15.9. Commutative, associative and Lie operads
Notes
Chapter 16. Invariant Birkhoff algebra and invariant Tits algebra
16.1. Invariant Birkhoff algebra
16.2. Invariant Tits algebra
16.3. Solomon descent algebra
16.4. Enumeration of face-types
16.5. Structure constants of the invariant Tits algebra
16.6. Invariant Lie and Zie elements
16.7. Invariant lune-incidence algebra
16.8. Invariant Eulerian idempotents
16.9. Peirce decompositions
16.10. Bilinear forms
16.11. Garsia-Reutenauer idempotents (Type ?)
16.12. Bergeron idempotents (Type ?)
Notes
Appendices
Appendix A. Regular cell complexes
A.1. Cell complexes
A.2. Minimal galleries and gate property
Notes
Appendix B. Posets
B.1. Poset terminology
B.2. Graded posets
B.3. Semimodularity and join-distributivity
B.4. Strongly connected posets
B.5. Adjunctions between posets
Notes
Appendix C. Incidence algebras of posets
C.1. Incidence algebras and Möbius functions
C.2. Radical of an incidence algebra
C.3. Reduced incidence algebras
C.4. Poset cocycles and deformations of incidence algebras
Notes
Appendix D. Algebras and modules
D.1. Modules
D.2. Idempotents and nilpotents
D.3. Split-semisimple commutative algebras
D.4. Diagonalizability and Jordan-Chevalley decomposition
D.5. Radical, socle and semisimplicity
D.6. Invertible elements and zero divisors
D.7. Lifting idempotents
D.8. Elementary algebras
D.9. Algebra of a finite lattice
Notes
Appendix E. Bands
E.1. Bands
E.2. Distance functions
Notes
References
Bibliography
Notation Index
Subject Index
Back Cover
