Topics in Hyperplane Arrangements

Description

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 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, 𝑊-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 𝐵 6.8. Arrangement of type 𝐷 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 𝐵 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 𝐴 Eulerian idempotents and Adams elements 12.6. Type 𝐵 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 𝐴) Lie elements 14.10. Type 𝐵 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 𝐴) 16.12. Bergeron idempotents (Type 𝐵) 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