Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting...

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Bibliographic Details
Main Authors: Margolis, Stuart, Saliola, Franco V., Steinberg, Benjamin
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2022
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Associative rings and algebras > Homological methods > Homological dimension. msc
Associative rings and algebras > Representation theory of rings and algebras > Representations of Artinian rings. msc
Associative rings and algebras > Rings and algebras arising under various constructions > Quadratic and Koszul algebras. msc
Combinatorial analysis
Combinatorial geometry
Combinatorics > Algebraic combinatorics > Combinatorial aspects of representation theory. msc
Convex and discrete geometry > Discrete geometry > Arrangements of points, flats, hyperplanes. msc
Convex and discrete geometry > Discrete geometry > Oriented matroids. msc
Convex and discrete geometry > Polytopes and polyhedra > Combinatorial properties (number of faces, shortest paths, etc.). msc
CW complexes
Group theory and generalizations > Semigroups > Representation of semigroups; actions of semigroups on sets. msc
Group theory and generalizations > Semigroups > Semigroup rings, multiplicative semigroups of rings. msc
Partially ordered sets
Representations of algebras
Semigroups
quivers
zonotopes
CAT zonotopal complexes
finite dimensional algebras
CAT cube complexes
hyperplane arrangements
Regular cell complexes
left regular bands
oriented matroids
ISBN:9781470450427, 1470450429
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Table of Contents:
  • Preface -- Acknowledgements -- Introduction -- Left Regular Bands, Hyperplane Arrangements, Oriented Matroids and Generalizations -- Regular CW Complexes and CW Posets -- Algebras -- Projective Resolutions and Global Dimension -- Quiver Presentations -- Quadratic and Koszul Duals -- Injective Envelopes for Hyperplane Arrangements, Oriented Matroids, CAT(0) Cube Complexes and COMs -- Enumeration of Cells for CW Left Regular Bands -- Cohomological Dimension
  • 10.1. Cohomology of left regular bands -- Bibliography -- Nomenclature -- Index -- Back Cover
  • 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