Topics in Hyperplane Arrangements
Marcelo Aguiar · Swapneel Mahajan
āļ.āļĒ. 2017 · Mathematical Surveys and Monographs āļŦāļāļąāļāļŠāļ·āļāđāļĨāđāļĄāļāļĩāđ 226 · American Mathematical Soc.
eBook
611
āļŦāļāđāļē
reportāļāļ°āđāļāļāđāļĨāļ°āļĢāļĩāļ§āļīāļ§āđāļĄāđāđāļāđāļĢāļąāļāļāļēāļĢāļāļĢāļ§āļāļŠāļāļāļĒāļ·āļāļĒāļąāļ Â āļāļđāļāđāļāļĄāļđāļĨāđāļāļīāđāļĄāđāļāļīāļĄ
āđāļāļĩāđāļĒāļ§āļāļąāļ eBook āđāļĨāđāļĄāļāļĩāđ
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 MÃķbius 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.
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 MÃķbius 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.
āđāļāļĩāđāļĒāļ§āļāļąāļāļāļđāđāđāļāđāļ
Marcelo Aguiar: Cornell University, Ithaca, NY,
Swapneel Mahajan: Indian Institute of Technology(IIT), Mumbai, India
āđāļŦāđāļāļ°āđāļāļ eBook āļāļĩāđ
āđāļŠāļāļāļāļ§āļēāļĄāđāļŦāđāļāļāļāļāļāļļāļāđāļŦāđāđāļĢāļēāļĢāļąāļāļĢāļđāđ
āļāđāļāļĄāļđāļĨāđāļāļāļēāļĢāļāđāļēāļ
āļŠāļĄāļēāļĢāđāļāđāļāļāđāļĨāļ°āđāļāđāļāđāļĨāđāļ
āļāļīāļāļāļąāđāļāđāļāļ Google Play Books āļŠāļģāļŦāļĢāļąāļ Android āđāļĨāļ° iPad/iPhone āđāļāļāļāļ°āļāļīāļāļāđāđāļāļĒāļāļąāļāđāļāļĄāļąāļāļīāļāļąāļāļāļąāļāļāļĩāļāļāļāļāļļāļ āđāļĨāļ°āļāđāļ§āļĒāđāļŦāđāļāļļāļāļāđāļēāļāđāļāļāļāļāļāđāļĨāļāđāļŦāļĢāļ·āļāļāļāļāđāļĨāļāđāđāļāđāļāļļāļāļāļĩāđ
āđāļĨāđāļāļāđāļāļāđāļĨāļ°āļāļāļĄāļāļīāļ§āđāļāļāļĢāđ
āļāļļāļāļāļąāļāļŦāļāļąāļāļŠāļ·āļāđāļŠāļĩāļĒāļāļāļĩāđāļāļ·āđāļāļāļēāļ Google Play āđāļāļĒāđāļāđāđāļ§āđāļāđāļāļĢāļēāļ§āđāđāļāļāļĢāđāđāļāļāļāļĄāļāļīāļ§āđāļāļāļĢāđāđāļāđ
eReader āđāļĨāļ°āļāļļāļāļāļĢāļāđāļāļ·āđāļāđ
āļŦāļēāļāļāđāļāļāļāļēāļĢāļāđāļēāļāļāļāļāļļāļāļāļĢāļāđ e-ink āđāļāđāļ Kobo eReader āļāļļāļāļāļ°āļāđāļāļāļāļēāļ§āļāđāđāļŦāļĨāļāđāļĨāļ°āđāļāļāđāļāļĨāđāđāļāļĒāļąāļāļāļļāļāļāļĢāļāđāļāļāļāļāļļāļ āđāļāļĢāļāļāļģāļāļēāļĄāļ§āļīāļāļĩāļāļēāļĢāļāļĒāđāļēāļāļĨāļ°āđāļāļĩāļĒāļāđāļāļĻāļđāļāļĒāđāļāđāļ§āļĒāđāļŦāļĨāļ·āļāđāļāļ·āđāļāđāļāļāđāļāļĨāđāđāļāļĒāļąāļ eReader āļāļĩāđāļĢāļāļāļĢāļąāļ
