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Author: Anders Bjorner
Publisher: Springer Science & Business Media
Published: 2006-02-25
Total Pages: 371
ISBN-13: 3540275967
DOWNLOAD EBOOKIncludes a rich variety of exercises to accompany the exposition of Coxeter groups Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups
Author: James E. Humphreys
Publisher: Cambridge University Press
Published: 1992-10
Total Pages: 222
ISBN-13: 9780521436137
DOWNLOAD EBOOKThis 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.
Author: Michael Davis
Publisher: Princeton University Press
Published: 2008
Total Pages: 601
ISBN-13: 0691131384
DOWNLOAD EBOOKThe 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.
Author: Drew Armstrong
Publisher: American Mathematical Soc.
Published: 2009-10-08
Total Pages: 176
ISBN-13: 0821844903
DOWNLOAD EBOOKThis memoir is a refinement of the author's PhD thesis -- written at Cornell University (2006). It is primarily a desription of new research but also includes a substantial amount of background material. At the heart of the memoir the author introduces and studies a poset $NC^{(k)}(W)$ for each finite Coxeter group $W$ and each positive integer $k$. When $k=1$, his definition coincides with the generalized noncrossing partitions introduced by Brady and Watt in $K(\pi, 1)$'s for Artin groups of finite type and Bessis in The dual braid monoid. When $W$ is the symmetric group, the author obtains the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman in Chain enumeration and non-crossing partitions.
Author: Francois Bergeron
Publisher: CRC Press
Published: 2009-07-06
Total Pages: 227
ISBN-13: 1439865078
DOWNLOAD EBOOKWritten for graduate students in mathematics or non-specialist mathematicians who wish to learn the basics about some of the most important current research in the field, this book provides an intensive, yet accessible, introduction to the subject of algebraic combinatorics. After recalling basic notions of combinatorics, representation theory, and
Author: Alexandre V. Borovik
Publisher: Springer Science & Business Media
Published: 2003-07-11
Total Pages: 292
ISBN-13: 9780817637644
DOWNLOAD EBOOKMatroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry, and "Coxeter Matroids" provides an intuitive and interdisciplinary treatment of their theory. In this text, matroids are examined in terms of symmetric and finite reflection groups; also, symplectic matroids and the more general coxeter matroids are carefully developed. The Gelfand-Serganova theorem, which allows for the geometric interpretation of matroids as convex polytopes with certain symmetry properties, is presented, and in the final chapter, matroid representations and combinatorial flag varieties are discussed. With its excellent bibliography and index and ample references to current research, this work will be useful for graduate students and research mathematicians.
Author: R. M. Green
Publisher: Cambridge University Press
Published: 2013-02-21
Total Pages: 329
ISBN-13: 1107026245
DOWNLOAD EBOOKUses the combinatorics and representation theory to construct and study important families of Lie algebras and Weyl groups.
Author: Anne Thomas
Publisher:
Published: 2018
Total Pages:
ISBN-13: 9783037191897
DOWNLOAD EBOOKAuthor: Howard Hiller
Publisher: Pitman Publishing
Published: 1982
Total Pages: 230
ISBN-13:
DOWNLOAD EBOOKAuthor: Daniel Bump
Publisher: World Scientific Publishing Company
Published: 2017-01-17
Total Pages: 292
ISBN-13: 9814733466
DOWNLOAD EBOOKThis unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.
