| Title | Reflection Groups and Coxeter Groups |
| Author(s) | James E. Humphreys |
| Publisher | Cambridge University Press |
| Year | 1992 |
| Edition | 1st edition |
| Language | English |
| Pages | 204 pages |
| ISBN | 9780521436137 |
| Genre / Domain | Mathematics, Algebra, Lie Theory |
| Series | Cambridge Studies in Advanced Mathematics |
| Size | 1.7 MB |
| Extension | DJVU |
Summary
"Reflection Groups and Coxeter Groups" by James E. Humphreys is a classic graduate textbook that provides a concrete and accessible introduction to the theory of Coxeter groups. Published by Cambridge University Press in 1992 as part of the Cambridge Studies in Advanced Mathematics series, this book has become a standard reference for students and researchers in algebra, geometry, and Lie theory. The text is self-contained, making it suitable for graduate courses, seminars, or self-study, and assumes only a foundational knowledge of abstract algebra and linear algebra.
The book is structured into two main parts, each serving a distinct pedagogical purpose. The first part is devoted to establishing concrete examples, beginning with finite reflection groups acting on Euclidean spaces. This includes a detailed discussion of root systems, crystallographic groups, and the classification of finite Coxeter groups. The first part culminates in the construction of affine Weyl groups, a class of Coxeter groups that plays a central role in Lie theory and representation theory. These concrete examples provide readers with a solid intuitive foundation before moving to more abstract concepts.
The second part of the book develops the general theory of Coxeter groups from scratch. It covers essential topics such as the Bruhat ordering, a partial order on the elements of a Coxeter group that is fundamental to the study of Schubert varieties and representation theory. The book also introduces the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups, a topic that has had a profound impact on modern representation theory. Throughout this section, Humphreys maintains a clear and rigorous exposition, providing proofs and examples that illuminate the abstract theory.
For readers, this book offers a comprehensive and well-organized treatment of a subject that is central to several areas of mathematics. It is particularly valuable for graduate students specializing in algebra, algebraic geometry, or Lie theory, as it provides the foundational knowledge needed for more advanced research. The book also includes a number of complementary topics and connections with Lie theory, sketched in later chapters, which give readers a sense of the broader mathematical landscape. An extensive bibliography on Coxeter groups and their applications further enhances the book's value as a research resource.
James E. Humphreys was a renowned mathematician and educator, and this book reflects his deep expertise and his commitment to clear exposition. His other works, including "Introduction to Lie Algebras and Representation Theory," are also highly regarded. "Reflection Groups and Coxeter Groups" remains a definitive and widely cited text in the field, appreciated for its balance of concrete examples and abstract theory. It is an essential addition to the library of any serious student or researcher in algebra and related fields.
Key Features
- Provides a concrete and accessible introduction to the theory of Coxeter groups.
- Self-contained text suitable for graduate courses, seminars, or self-study.
- Part one establishes concrete examples, including finite reflection groups and affine Weyl groups.
- Part two develops the general theory of Coxeter groups from scratch.
- Covers the Bruhat ordering and its applications in representation theory.
- Introduces the seminal work of Kazhdan and Lusztig on Hecke algebras.
- Includes connections with Lie theory and other complementary topics.
- Features an extensive bibliography on Coxeter groups and their applications.
- Published by Cambridge University Press in the Cambridge Studies in Advanced Mathematics series.
- Written by James E. Humphreys, a renowned expert in Lie theory and algebraic groups.
About Author
James Edward Humphreys (December 10, 1939 – August 27, 2020) was an American mathematician who specialized in algebraic groups, Lie groups, and Lie algebras, as well as their applications. He earned his Ph.D. from Yale University in 1966 and held academic positions at several institutions, including the University of Massachusetts Amherst and the University of Oregon. He was a prolific author of mathematical texts and is best known for his books "Introduction to Lie Algebras and Representation Theory" and "Reflection Groups and Coxeter Groups," which are widely used in graduate courses around the world.
Humphreys' work has had a significant impact on the fields of algebra and representation theory. His textbooks are praised for their clarity, rigor, and pedagogical effectiveness. He was a dedicated teacher and mentor, and his contributions to mathematics education have been recognized by the mathematical community. His legacy continues through his influential writings and the many students and researchers he inspired.
Related Books
- Introduction to Lie Algebras and Representation Theory — James E. Humphreys
- Lie Groups, Lie Algebras, and Representations: An Elementary Introduction — Brian Hall
- Coxeter Groups and Hecke Algebras — Andrew Mathas
- Buildings and Classical Groups — Paul B. Garrett
- Representation Theory: A First Course — William Fulton and Joe Harris
- Lie Algebras and Lie Groups — Jean-Pierre Serre
- Theory of Lie Groups — Claude Chevalley
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FAQ
Q : Who is the author of this book?
R : The author is James E. Humphreys, a renowned American mathematician who specialized in algebraic groups, Lie groups, and Lie algebras.
Q : What is the main focus of "Reflection Groups and Coxeter Groups"?
R : The book provides a comprehensive introduction to the theory of Coxeter groups, covering concrete examples, general theory, Bruhat ordering, and Hecke algebras.
Q : Who is the target audience for this book?
R : Graduate students and researchers in mathematics, particularly those specializing in algebra, geometry, and Lie theory.
Q : What are the two main parts of the book?
R : Part one establishes concrete examples, including finite reflection groups and affine Weyl groups. Part two develops the general theory of Coxeter groups from scratch.
Q : What is the ISBN of the book?
R : The ISBN is 9780521436137.
Q : What series is this book part of?
R : It is part of the Cambridge Studies in Advanced Mathematics series.
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