Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in Algebraic Combinatorics and Discrete Geometry.

Cover
Title page
Preface
Acknowledgements
Chapter 1. Introduction
Chapter 2. Left Regular Bands, Hyperplane Arrangements, Oriented Matroids and Generalizations
2.1. Green's relations and the structure of left regular bands
2.2. Free left regular bands and matroids
2.3. Free partially commutative left regular bands
2.4. Hyperplane arrangements and oriented matroids
2.5. Strong elimination systems, lopsided systems and COMs
2.6. Complex hyperplane arrangements
Chapter 3. Regular CW Complexes and CW Posets
3.1. Simplicial complexes and order complexes of posets
3.2. Regular CW complexes and CW posets
3.3. Oriented interval greedoids
3.4. The topology of left regular bands
3.5. CAT(0) cube complexes
3.6. CAT(0) zonotopal complexes
Chapter 4. Algebras
4.1. Rings and radicals
4.2. Finite dimensional algebras
4.3. Quivers and basic algebras
4.4. Gradings, quadratic algebras and Koszul algebras
4.5. The algebra of a left regular band
4.6. Existence of identity elements in left regular band algebras
4.7. Cartan invariants
Chapter 5. Projective Resolutions and Global Dimension
5.1. Actions of left regular bands on CW posets
5.2. Projective resolutions
5.3. Ext and global dimension
5.4. Minimal projective resolutions
Chapter 6. Quiver Presentations
6.1. A general result
6.2. A quiver presentation for CW left regular bands
Chapter 7. Quadratic and Koszul Duals
7.1. Quadratic duals
7.2. Koszul duals
Chapter 8. Injective Envelopes for Hyperplane Arrangements, Oriented Matroids, CAT(0) Cube Complexes and COMs
8.1. Generalities
8.2. Hyperplane arrangements
8.3. Oriented matroids
Chapter 9. Enumeration of Cells for CW Left Regular Bands
9.1. Flag vectors
9.2. Cartan invariants
Chapter 10. Cohomological Dimension.

10.1. Cohomology of left regular bands
Bibliography
Nomenclature
Index
Back Cover.