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AuthorBorovik, Alexandre V. author
TitleCoxeter Matroids [electronic resource] / by Alexandre V. Borovik, I. M. Gelfand, Neil White
ImprintBoston, MA : Birkhรคuser Boston, 2003
Connect tohttp://dx.doi.org/10.1007/978-1-4612-2066-4
Descript XXII, 266 p. online resource

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


CONTENT

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 Jordanโ{128}{148}Hรถlder 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 Symmetric group as a Coxeter group -- 5.18 Exercises for Chapter 5 -- 6 Coxeter Matroids -- 6.1 Coxeter matroids -- 6.2 Root systems -- 6.3 The Gelfandโ{128}{148}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


Mathematics Algebra Algebraic geometry Combinatorics Mathematics Algebraic Geometry Mathematics general Algebra Combinatorics



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