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This graduate textbook presents a concrete and up-to-date introduction to the theory of Coxeter groups. The book is self-contained, making it suitable either for courses and seminars or for self-study. The first part is devoted to establishing concrete examples. Finite reflection groups acting on Euclidean spaces are discussed, and the first part ends with the construction of the affine Weyl groups, a class of Coxeter groups that plays a major role in Lie theory. The second part (which is logically independent of, but motivated by, the first) develops from scratch the properties of Coxeter groups in general, including the Bruhat ordering and the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups is introduced. Finally a number of interesting complementary topics as well as connections with Lie theory are sketched. The book concludes with an extensive bibliography on Coxeter groups and their applications.
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Coxeter groups --- Reflection groups --- Groupes de Coxeter --- Coxeter's groups --- Real reflection groups --- Reflection groups, Real --- Coxeter groups. --- Reflection groups. --- 512.81 --- 512.81 Lie groups --- Lie groups --- Finite groups --- Transformations (Mathematics) --- Group theory --- #KOPO:Prof. R. Holvoet
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The book is the first to give a comprehensive overview of the techniques and tools currently being used in the study of combinatorial problems in Coxeter groups. It is self-contained, and accessible even to advanced undergraduate students of mathematics. The primary purpose of the book is to highlight approximations to the difficult isomorphism problem in Coxeter groups. A number of theorems relating to this problem are stated and proven. Most of the results addressed here concern conditions which can be seen as varying degrees of uniqueness of representations of Coxeter groups. Throughout the
Coxeter groups. --- Isomorphisms (Mathematics) --- Categories (Mathematics) --- Group theory --- Morphisms (Mathematics) --- Set theory --- Coxeter's groups --- Real reflection groups --- Reflection groups, Real
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Coxeter groups are groups generated by reflections, and they appear throughout mathematics. Tits developed the general theory of Coxeter groups in order to develop the theory of buildings. Buildings have interrelated algebraic, combinatorial and geometric structures, and are powerful tools for understanding the groups which act on them. These notes focus on the geometry and topology of Coxeter groups and buildings, especially nonspherical cases. The emphasis is on geometric intuition, and there are many examples and illustrations. Part I describes Coxeter groups and their geometric realisations, particularly the Davis complex, and Part II gives a concise introduction to buildings. This book will be suitable for mathematics graduate students and researchers in geometric group theory, as well as algebra and combinatorics. The assumed background is basic group theory, including group actions, and basic algebraic topology, together with some knowledge of Riemannian geometry.
Coxeter groups. --- Geometric group theory. --- Coxeter's groups --- Real reflection groups --- Reflection groups, Real --- Group theory --- Geometric group theory --- Coxeter groups
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The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.
512.54 --- 512.54 Groups. Group theory --- Groups. Group theory --- Coxeter groups --- Geometric group theory --- Coxeter's groups --- Real reflection groups --- Reflection groups, Real --- Group theory --- Coxeter groups. --- Geometric group theory.
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This monograph provides a comprehensive introduction to the Kazhdan-Lusztig theory of cells in the broader context of the unequal parameter case. Serving as a useful reference, the present volume offers a synthesis of significant advances made since Lusztig’s seminal work on the subject was published in 2002. The focus lies on the combinatorics of the partition into cells for general Coxeter groups, with special attention given to induction methods, cellular maps and the role of Lusztig's conjectures. Using only algebraic and combinatorial methods, the author carefully develops proofs, discusses open conjectures, and presents recent research, including a chapter on the action of the cactus group. Kazhdan-Lusztig Cells with Unequal Parameters will appeal to graduate students and researchers interested in related subject areas, such as Lie theory, representation theory, and combinatorics of Coxeter groups. Useful examples and various exercises make this book suitable for self-study and use alongside lecture courses.
Mathematics. --- Group theory. --- Group Theory and Generalizations. --- Hecke algebras. --- Coxeter groups. --- Representations of algebras. --- Algebra --- Coxeter's groups --- Real reflection groups --- Reflection groups, Real --- Group theory --- Algebras, Hecke --- Group algebras --- Groups, Theory of --- Substitutions (Mathematics)
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Coxeter groups --- Hopf algebras --- 512.54 --- 512.54 Groups. Group theory --- Groups. Group theory --- Algebras, Hopf --- Algebraic topology --- Coxeter's groups --- Real reflection groups --- Reflection groups, Real --- Group theory
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Reflection Groups and their invariant theory provide the main themes of this book and the first two parts focus on these topics. The first 13 chapters deal with reflection groups (Coxeter groups and Weyl groups) in Euclidean Space while the next thirteen chapters study the invariant theory of pseudo-reflection groups. The third part of the book studies conjugacy classes of the elements in reflection and pseudo-reflection groups. The book has evolved from various graduate courses given by the author over the past 10 years. It is intended to be a graduate text, accessible to students with a basic background in algebra. Richard Kane is a professor of mathematics at the University of Western Ontario. His research interests are algebra and algebraic topology. Professor Kane is a former President of the Canadian Mathematical Society.
Group theory --- Reflection groups --- Invariants --- Mathematical analysis. --- Analysis (Mathematics). --- Geometry. --- Analysis. --- Mathematics --- Euclid's Elements --- 517.1 Mathematical analysis --- Mathematical analysis
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Reflection groups --- Braid theory --- Hecke algebras --- Finite groups --- Groupes de réflexions (mathématiques) --- Tresses, Théorie des --- Hecke, Algèbres de --- Groupes finis --- Tresses, Théorie des. --- Hecke, Algèbres de. --- Groupes finis.
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