Coxeter Matroids

Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry. This largely self-contained text provides an intuitive and interdisciplinary treatment of Coxeter matroids, a new and beautiful generalization of matroids which is based on a finite Coxeter group.
Key topics and features:
* Systematic, clearly written exposition with ample references to current research
* Matroids are examined in terms of symmetric and finite reflection groups
* Finite reflection groups and Coxeter groups are developed from scratch
* The Gelfand-Serganova theorem is presented, allowing for a geometric interpretation of matroids and Coxeter matroids as convex polytopes with certain symmetry properties
* Matroid representations in buildings and combinatorial flag varieties are studied in the final chapter
* Many exercises throughout
* Excellent bibliography and index
Accessible to graduate students and research mathematicians alike, "Coxeter Matroids" can be used as an introductory survey, a graduate course text, or a reference volume.

List of contents

1 Matroids and Flag Matroids.- 1.1 Matroids.- 1.2 Representable matroids.- 1.3 Maximality Property.- 1.4 Increasing Exchange Property.- 1.5 Sufficient systems of exchanges.- 1.6 Matroids as maps.- 1.7 Flag matroids.- 1.8 Flag matroids as maps.- 1.9 Exchange properties for flag matroids.- 1.10 Root system.- 1.11 Polytopes associated with flag matroids.- 1.12 Properties of matroid polytopes.- 1.13 Minkowski sums.- 1.14 Exercises for Chapter 1.- 2 Matroids and Semimodular Lattices.- 2.1 Lattices as generalizations of projective geometry.- 2.2 Semimodular lattices.- 2.3 Jordan-Hölder permutation.- 2.4 Geometric lattices.- 2.5 Representations of matroids.- 2.6 Representation of flag matroids.- 2.7 Every flag matroid is representable.- 2.8 Exercises for Chapter 2.- 3 Symplectic Matroids.- 3.1 Definition of symplectic matroids.- 3.2 Root systems of type Cn.- 3.3 Polytopes associated with symplectic matroids.- 3.4 Representable symplectic matroids.- 3.5 Homogeneous symplectic matroids.- 3.6 Symplectic flag matroids.- 3.7 Greedy Algorithm.- 3.8 Independent sets.- 3.9 Symplectic matroid constructions.- 3.10 Orthogonal matroids.- 3.11 Open problems.- 3.12 Exercises for Chapter 3.- 4 Lagrangian Matroids.- 4.1 Lagrangian matroids.- 4.2 Circuits and strong exchange.- 4.3 Maps on orientable surfaces.- 4.4 Exercises for Chapter 4.- 5 Reflection Groups and Coxeter Groups.- 5.1 Hyperplane arrangements.- 5.2 Polyhedra and polytopes.- 5.3 Mirrors and reflections.- 5.4 Root systems.- 5.5 Isotropy groups.- 5.6 Parabolic subgroups.- 5.7 Coxeter complex.- 5.8 Labeling of the Coxeter complex.- 5.9 Galleries.- 5.10 Generators and relations.- 5.11 Convexity.- 5.12 Residues.- 5.13 Foldings.- 5.14 Bruhat order.- 5.15 Splitting the Bruhat order.- 5.16 Generalized permutahedra.- 5.17 Symmetricgroup as a Coxeter group.- 5.18 Exercises for Chapter 5.- 6 Coxeter Matroids.- 6.1 Coxeter matroids.- 6.2 Root systems.- 6.3 The Gelfand-Serganova Theorem.- 6.4 Coxeter matroids and polytopes.- 6.5 Examples.- 6.6 W-matroids.- 6.7 Characterization of matroid maps.- 6.8 Adjacency in matroid polytopes.- 6.9 Combinatorial adjacency.- 6.10 The matroid polytope.- 6.11 Exchange groups of Coxeter matroids.- 6.12 Flag matroids and concordance.- 6.13 Combinatorial flag variety.- 6.14 Shellable simplicial complexes.- 6.15 Shellability of the combinatorial flag variety.- 6.16 Open problems.- 6.17 Exercises for Chapter 6.- 7 Buildings.- 7.1 Gaussian decomposition.- 7.2 BN-pairs.- 7.3 Deletion Property.- 7.4 Deletion property and Coxeter groups.- 7.5 Reflection representation of W.- 7.6 Classification of finite Coxeter groups.- 7.7 Chamber systems.- 7.8 W-metric.- 7.9 Buildings.- 7.10 Representing Coxeter matroids in buildings.- 7.11 Vector-space representations and building representations.- 7.12 Residues in buildings.- 7.13 Buildings of type An-1 = Symn.- 7.14 Combinatorial flag varieties, revisited.- 7.15 Open Problems.- 7.16 Exercises for Chapter 7.- References.

About the author

Neil White, Journalist und ehemaliger Verleger von Hochglanzmagazinen (New Orleans Magazine, Coast Magazine, Coast Business Journal), lebt heute als Autor von Theaterstücken und Essays in Oxford, Mississippi, wo er einen kleinen Verlag besitzt.

Summary

Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry. This largely self-contained text provides an intuitive and interdisciplinary treatment of Coxeter matroids, a new and beautiful generalization of matroids which is based on a finite Coxeter group.

Key topics and features:

* Systematic, clearly written exposition with ample references to current research
* Matroids are examined in terms of symmetric and finite reflection groups
* Finite reflection groups and Coxeter groups are developed from scratch
* The Gelfand-Serganova theorem is presented, allowing for a geometric interpretation of matroids and Coxeter matroids as convex polytopes with certain symmetry properties
* Matroid representations in buildings and combinatorial flag varieties are studied in the final chapter
* Many exercises throughout
* Excellent bibliography and index

Accessible to graduate students and research mathematicians alike, "Coxeter Matroids" can be used as an introductory survey, a graduate course text, or a reference volume.

Additional text

From the reviews:

"This largely self-contained text provides an intuitive and interdisciplinary treatment of Coxeter matroids, a new and beautiful generalization of matroids which is based on a finite Coxeter group."

— ZENTRALBLATT MATH

"...this accessible and well-written book, intended to be "a cross between a postgraduate text and a research monograph," is well worth reading and makes a good case for doing matroids with mirrors."

— SIAM REVIEW

"This accessible and well-written book, intended to be ‘a cross between a postgraduate text and a research monograph,’ is well worth reading and makes a good case for doing matroids with mirrors." (Joseph Kung, SIAM Review, Vol. 46 (3), 2004)

"This accessible and well-written book, designed to be ‘a cross between a postgraduate text and a research monograph’, should win many converts.”(MATHEMATICAL REVIEWS)

Report

From the reviews:
"This largely self-contained text provides an intuitive and interdisciplinary treatment of Coxeter matroids, a new and beautiful generalization of matroids which is based on a finite Coxeter group."
- ZENTRALBLATT MATH
"...this accessible and well-written book, intended to be "a cross between a postgraduate text and a research monograph," is well worth reading and makes a good case for doing matroids with mirrors."
- SIAM REVIEW
"This accessible and well-written book, intended to be 'a cross between a postgraduate text and a research monograph,' is well worth reading and makes a good case for doing matroids with mirrors." (Joseph Kung, SIAM Review, Vol. 46 (3), 2004)
"This accessible and well-written book, designed to be 'a cross between a postgraduate text and a research monograph', should win many converts."(MATHEMATICAL REVIEWS)

Product details

• Authors: Borovik, Alexandre Borovik, Alexandre V Borovik, Alexandre V. Borovik, I. M. Gelfand, Israel Gelfand, Israel M Gelfand, Israel M. Gelfand, Izrail M. Gelfand, M. I. Gelfand, Neil White
• Assisted by: A. Borovik (Illustration), Alexandre V. Borovik (Illustration), Israel M. Gelfand (Illustration)
• Publisher: Springer, Basel
• Languages: English
• Product format: Hardback
• Released: 08.08.2003
• EAN: 9780817637644
• ISBN: 978-0-8176-3764-4
• No. of pages: 266
• Weight: 617 g
• Illustrations: XXII, 266 p.
• Series: Progress in Mathematics; Progress in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Geometry; Algebra, C, Mathematics, Combinatorics, Mathematics and Statistics, Mathematics, general, Algebraic Geometry, Discrete Mathematics, Combinatorics & graph theory