Combinatorics Of Coxeter Groups
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Combinatorics Of Coxeter Groups
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Author : Anders Bjorner
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-02-25
Combinatorics Of Coxeter Groups written by Anders Bjorner and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-02-25 with Mathematics categories.
Includes 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
Generalized Noncrossing Partitions And Combinatorics Of Coxeter Groups
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Author : Drew Armstrong
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-10-08
Generalized Noncrossing Partitions And Combinatorics Of Coxeter Groups written by Drew Armstrong and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-10-08 with Mathematics categories.
This 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.
Combinatorics And Topology Related To Involutions In Coxeter Groups
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Author : Mikael Hansson
language : en
Publisher: Linköping University Electronic Press
Release Date : 2018-05-21
Combinatorics And Topology Related To Involutions In Coxeter Groups written by Mikael Hansson and has been published by Linköping University Electronic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-05-21 with categories.
This dissertation consists of three papers in combinatorial Coxeter group theory. A Coxeter group is a group W generated by a set S, where all relations can be derived from the relations s2 = e for all s ?? S, and (ss?)m(s,s?) = e for some pairs of generators s ? s? in S, where e ?? W is the identity element and m(s, s?) is an integer satisfying that m(s, s?) = m(s?, s) ? 2. Two prominent examples of Coxeter groups are provided by the symmetric group Sn (i.e., the set of permutations of {1, 2, . . . , n}) and finite reflection groups (i.e., finite groups generated by reflections in some real euclidean space). There are also important infinite Coxeter groups, e.g., affine reflection groups. Every Coxeter group can be equipped with various natural partial orders, the most important of which is the Bruhat order. Any subset of a Coxeter group can then be viewed as an induced subposet. In Paper A, we study certain posets of this kind, namely, unions of conjugacy classes of involutions in the symmetric group. We obtain a complete classification of the posets that are pure (i.e., all maximal chains have the same length). In particular, we prove that the set of involutions with exactly one fixed point is pure, which settles a conjecture of Hultman in the affirmative. When the posets are pure, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, established by Can, Cherniavsky, and Twelbeck. Paper B also deals with involutions in Coxeter groups. Given an involutive automorphism ? of a Coxeter system (W, S), let ?(?) = {w ?? W | ?(w) = w?1} be the set of twisted involutions. In particular, ?(id) is the set of ordinary involutions in W. It is known that twisted involutions can be represented by words in the alphabet = { | s ?? S}, called -expressions. If ss? has finite order m(s, s?), let a braid move be the replacement of ? ? by ? ? ?, both consisting of m(s, s?) letters. We prove a word property for ?(?), for any Coxeter system (W, S) with any ?. More precisely, we provide a minimal set of moves, easily determined from the Coxeter graph of (W, S), that can be added to the braid moves in order to connect all reduced -expressions for any given w ?? ?(?). This improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg. In Paper C, we investigate the topology of (the order complexes of) certain posets, called pircons. A special partial matching (SPM) on a poset is a matching of the Hasse diagram satisfying certain extra conditions. An SPM without fixed points is precisely a special matching as defined by Brenti. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti’s zircons. Our main result is that every open interval in a pircon is a PL ball or a PL sphere. An important subset of ?(?) is the set ??(?) = {?(w?1)w | w ?? W} of twisted identities. We prove that if ? does not flip any edges with odd labels in the Coxeter graph, then ??(?), with the order induced by the Bruhat order on W, is a pircon. Hence, its open intervals are PL balls or spheres, which confirms a conjecture of Hultman. It is also demonstrated that Bruhat orders on Rains and Vazirani’s quasiparabolic W-sets (under a boundedness assumption) form pircons. In particular, this applies to all parabolic quotients of Coxeter groups.
Coxeter Groups And Hopf Algebras
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Author : Marcelo Aguiar
language : en
Publisher: American Mathematical Soc.
Release Date : 2006
Coxeter Groups And Hopf Algebras written by Marcelo Aguiar and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Education categories.
An important idea in the work of G.-C. Rota is that certain combinatorial objects give rise to Hopf algebras that reflect the manner in which these objects compose and decompose. Recent work has seen the emergence of several interesting Hopf algebras of this kind, which connect diverse subjects such as combinatorics, algebra, geometry, and theoretical physics. This monograph presents a novel geometric approach using Coxeter complexes and the projection maps of Tits for constructing and studying many of these objects as well as new ones. The first three chapters introduce the necessary background ideas making this work accessible to advanced graduate students. The later chapters culminate in a unified and conceptual construction of several Hopf algebras based on combinatorial objects which emerge naturally from the geometric viewpoint. This work lays a foundation and provides new insights for further development of the subject.
The Isomorphism Problem In Coxeter Groups
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Author : C. Patrick Bahls
language : en
Publisher: World Scientific
Release Date : 2005
The Isomorphism Problem In Coxeter Groups written by C. Patrick Bahls and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.
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 investigation, the readers are introduced to a large number of tools in the theory of Coxeter groups, drawn from dozens of recent articles by prominent researchers in geometric and combinatorial group theory, among other fields. As the central problem of the book may in fact be solved soon, the book aims to go further, providing the readers with many techniques that can be used to answer more general questions. The readers are challenged to practice those techniques by solving exercises, a list of which concludes each chapter.
The Coxeter Legacy
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Author : Harold Scott Macdonald Coxeter
language : en
Publisher: American Mathematical Soc.
Release Date :
The Coxeter Legacy written by Harold Scott Macdonald Coxeter and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.
This collection of essays on the legacy of mathematican Donald Coxeter is a mixture of surveys, updates, history, storytelling and personal memories covering both applied and abstract maths. Subjects include: polytopes, Coxeter groups, equivelar polyhedra, Ceva's theorum, and Coxeter and the artists.
Coxeter Matroids
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Author : Alexandre V. Borovik
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Coxeter Matroids written by Alexandre V. Borovik and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Matroids 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.
Combinatorics Of Minuscule Representations
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Author : R. M. Green
language : en
Publisher: Cambridge University Press
Release Date : 2013-02-21
Combinatorics Of Minuscule Representations written by R. M. Green and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-02-21 with Mathematics categories.
Uses the combinatorics and representation theory to construct and study important families of Lie algebras and Weyl groups.
Reflection Groups And Coxeter Groups
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Author : James E. Humphreys
language : en
Publisher: Cambridge University Press
Release Date : 1992-10
Reflection Groups And Coxeter Groups written by James E. Humphreys and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-10 with Mathematics categories.
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.
Algebraic Combinatorics And Coinvariant Spaces
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Author : Francois Bergeron
language : en
Publisher: CRC Press
Release Date : 2009-07-06
Algebraic Combinatorics And Coinvariant Spaces written by Francois Bergeron and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-07-06 with Mathematics categories.
Written 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