Topics in Hyperplane Arrangements
About this Title
Marcelo Aguiar, Cornell University, Ithaca, NY and Swapneel Mahajan, Indian Institute of Technology(IIT), Mumbai, India
Publication: Mathematical Surveys and Monographs
Publication Year: 2017; Volume 226
ISBNs: 978-1-4704-3711-4 (print); 978-1-4704-4254-5 (online)
MathSciNet review: MR3726871
MSC: Primary 52C35; Secondary 05E15, 06A07, 06C10, 17B01, 18G35, 20F55
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.
Graduate students and researchers interested in hyperplane arrangements (of interest in several areas of mathematics).
Table of Contents
- Hyperplane arrangements
- Category of lunes
- Reflection arrangements
- Braid arrangement and related examples
- Descent and lune equations
- Distance functions and Varchenko matrix
- Birkhoff algebra and Tits algebra
- Lie and Zie elements
- Eulerian idempotents
- Diagonalizability and characteristic elements
- Loewy series and Peirce decompositions
- Dynkin idempotents
- Incidence algebras
- Invariant Birkhoff algebra and invariant Tits algebra