Buch, Englisch, Format (B × H): 155 mm x 235 mm
Reihe: Algebra and Applications
ISBN: 978-3-032-45161-3
Verlag: Springer
This graduate text offers a guided tour through the remarkable world of exceptional finite complex reflection groups and their connections with algebraic geometry. Drawing on decades of research, the author develops a self-contained account that combines group theory, invariant theory, and geometry, emphasizing explicit constructions, concrete examples, and exceptional objects that illuminate the richness of the subject. After a concise review of the necessary background and the fundamental results on reflection groups and their invariants, the book focuses on the exceptional groups themselves. New and classical constructions are presented, together with the exceptional morphisms linking certain complex reflection groups to Coxeter groups. These methods provide fresh insight into the relationships between exceptional groups, their invariants, and the algebraic structures that surround them.
Throughout, the emphasis is on examples, computations, and geometric interpretations that complement standard references in the field. The book is prime for topics courses as well as advanced courses in finite reflection groups and invariant theory, to name just two. By bringing together material scattered across the literature and highlighting deep interactions between algebra and geometry through the notion of symmetry, this volume will be valuable to researchers in reflection groups and algebraic geometry, graduate students seeking rich examples, and mathematicians interested in exceptional phenomena.
Zielgruppe
Research
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
Preface.- List of word records.- Acknowledgements.- I Preliminaries.- 1 Notation, terminology.- 2 Basic invariant theory.- II Review of classical theory of reflection groups.- 3 Reflections, reflection groups.- 4 Classification.- 5 Reflection groups and invariant theory.- 6 Springer/Lehrer-Springer theory.- III Structure of reflection groups.- 7 Abelainness.- 8 Reflection quotients.- 9 Remarks about Sylow subgroups.- IV About exceptional groups.- 10 Exceptional morphisms.- 11 Complement: finite subgroups of SL2(C).- 12 Some reflection groups of rank 2.- 13 Some reflection groups of rank 3.- 14 Alternative construction of the two exceptional Coxeter groups of rank 4.- 15 The groups G32 ad W(B6), and the outer automorphism of G6.- 16 The groups G28 and G29 as reflection subgroups of G31.- 17 The groups G32 and G35 = W (E6).- 18 The groups G37 = W (E8), G31, G30= W(H4) and G22 altogether.- V Some exceptional groups and geometric configurations.- 19 Notation, terminology, preliminaries.- 20 The group G24, the Klein curve and a configuration of 7 conics.- 21 The groups G25 and G26 and the Hesse pencil.- 22. The group G27, Gerbaldi Theorem and Valentiner group.- 23 Surfaces with many lines or conics.- VI Singular curves and surfaces built from invariants.- 24 ADE singularities.- 25 Complements about \mu^{sur}_{A1}(d).- 26 Singular plane curves.- 27 The group G30 and surfaces with many nodes.- 28 Surfaces with many D4-singularities.- VII Appendices.- A Some facts finite groups.- B Recollection on Coxeter groups.- VIII Solutions to exercises.- Bibliography.
