Inhaltsangabe1 Matroids and Flag Matroids.- 1.1 Matroids.- 1.1.1 Definition in terms of bases.- 1.1.2 Examples.- 1.1.3 Circuits.- 1.2 Representable matroids.- 1.3 Maximality Property.- 1.4 Increasing Exchange Property.- 1.5 Sufficient systems of exchanges.- 1.5.1 Strong Exchange Property.- 1.6 Matroids as maps.- 1.7 Flag matroids.- 1.7.1 Flags.- 1.7.2 Flag matroids.- 1.7.3 Matroid quotients.- 1.7.4 Equivalence of Maximality Property and concordance of constituents.- 1.7.5 Representable flag matroids.- 1.7.6 Higgs lift.- 1.8 Flag matroids as maps.- 1.9 Exchange properties for flag matroids.- 1.9.1 Increasing Exchange Property for flag matroids.- 1.9.2 Failure of the Strong Exchange Property for flag matroids.- 1.10 Root system.- 1.10.1 Roots.- 1.10.2 Transpositions and reflections.- 1.10.3 Geometric representation of flags.- 1.10.4 Orderings associated with the root system.- 1.11 Polytopes associated with flag matroids.- 1.11.1 Polytopes associated with flag matroids.- 1.11.2 Main Theorem.- 1.12 Properties of matroid polytopes.- 1.12.1 Adjacency in matroids.- 1.12.2 Groups generated by transpositions.- 1.12.3 Components of matroids and the transposition graph.- 1.12.4 2-dimensional faces of matroid polytopes.- 1.12.5 Dimension of the matroid polytope.- 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.- 2.4.1 Bases of lattices.- 2.4.2 Closure operators.- 2.4.3 Geometric lattice determined by a matroid.- 2.5 Representations of matroids.- 2.6 Representation of flag matroids.- 2.6.1 Retractions.- 2.6.2 Matroid maps from chains.- 2.7 Every flag matroid is representable.- 2.8 Exercises for Chapter 2.- 3 Symplectic Matroids.- 3.1 Definition of symplectic matroids.- 3.1.1 Hyperoctahedral group and admissible permutations.- 3.1.2 Admissible orderings.- 3.1.3 Symplectic matroids.- 3.2 Root systems of type Cn.- 3.2.1 Roots.- 3.2.2 Simple systems of roots.- 3.2.3 Correspondences.- 3.3 Polytopes associated with symplectic matroids.- 3.3.1 Geometric representation of admissible sets.- 3.3.2 Gelfand-Serganova Theorem for symplectic matroids.- 3.4 Representable symplectic matroids.- 3.4.1 Isotropic subspaces.- 3.4.2 Symplectic matroids from isotropic subspaces.- 3.4.3 Examples.- 3.4.4 Operations on representations.- 3.5 Homogeneous symplectic matroids.- 3.6 Symplectic flag matroids.- 3.6.1 Examples.- 3.6.2 Representable symplectic flag matroids.- 3.7 Greedy Algorithm.- 3.8 Independent sets.- 3.9 Symplectic matroid constructions.- 3.10 Orthogonal matroids.- 3.10.1 Dn-admissible orderings.- 3.10.2 Orthogonal matroids.- 3.10.3 Representable orthogonal matroids.- 3.10.4 Orthogonal flag matroids.- 3.11 Open problems.- 3.12 Exercises for Chapter 3.- 4 Lagrangian Matroids.- 4.1 Lagrangian matroids.- 4.1.1 Transversals.- 4.1.2 Symmetric Exchange Axiom.- 4.1.3 Represented Lagrangian matroids.- 4.1.4 Homogeneous Lagrangian matroids.- 4.2 Circuits and strong exchange.- 4.2.1 Dual matroid.- 4.2.2 Circuits.- 4.2.3 Circuits and cocircuits.- 4.2.4 Strong Exchange Property.- 4.2.5 Circuit characterizations of Lagrangian matroids.- 4.3 Maps on orientable surfaces.- 4.3.1 Maps on compact surfaces.- 4.3.2 Matroids, representations and maps.- 4.4 Exercises for Chapter 4.- 5 Reflection Groups and Coxeter Groups.- 5.1 Hyperplane arrangements.- 5.1.1 Chambers of a hyperplane arrangement.- 5.1.2 Galleries.- 5.2 Polyhedra and polytopes.- 5.3 Mirrors and reflections.- 5.3.1 Systems of mirrors and of reflections.- 5.3.2 Finite reflection groups.- 5.4 Root systems.- 5.4.1 Mirrors and their normal vectors.- 5.4.2 Root systems.- 5.4.3 Positive and simple systems.- 5.4.4 Classification of root systems.- 5.5 Isotropy groups.- 5.6 Parabolic subgroups.- 5.7 Coxeter complex.- 5.7.1 Chambers.- 5.7.2 Generation by simple reflections.- 5.7.3 Action of W on W.- 5.8 Labeling of the Coxeter complex.- 5.9 Galleries.- 5.