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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| Hlavní autori: | , , |
|---|---|
| Médium: | E-kniha Kniha |
| Jazyk: | English |
| Vydavateľské údaje: |
Providence, Rhode Island
American Mathematical Society
2022
|
| Vydanie: | 1 |
| Edícia: | Memoirs of the American Mathematical Society |
| Predmet: |
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
Convex and discrete geometry
> Polytopes and polyhedra
> Combinatorial properties (number of faces, shortest paths, etc.). msc
Group theory and generalizations
> Semigroups
> Representation of semigroups; actions of semigroups on sets. msc
|
| ISBN: | 9781470450427, 1470450429 |
| ISSN: | 0065-9266, 1947-6221 |
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| Abstract | 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 Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left
regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such
structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left
regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order
complexes of posets naturally associated to the left regular band.
The purpose of the present monograph is to further develop and
deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all
simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left
regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the
examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure
on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional
oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A
fairly complete picture of the representation theory for CW left regular bands is obtained. |
|---|---|
| AbstractList | 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 Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left
regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such
structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left
regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order
complexes of posets naturally associated to the left regular band.
The purpose of the present monograph is to further develop and
deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all
simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left
regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the
examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure
on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional
oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A
fairly complete picture of the representation theory for CW left regular bands is obtained. View the abstract. |
| Author | Steinberg, Benjamin Saliola, Franco V. Margolis, Stuart |
| Author_xml | – sequence: 1 givenname: Stuart surname: Margolis fullname: Margolis, Stuart – sequence: 2 givenname: Franco V. surname: Saliola fullname: Saliola, Franco V. – sequence: 3 givenname: Benjamin surname: Steinberg fullname: Steinberg, Benjamin |
| BackLink | https://cir.nii.ac.jp/crid/1130573251094305170$$DView record in CiNii |
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| Copyright | Copyright 2021 American Mathematical Society |
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| DOI | 10.1090/memo/1345 |
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| Keywords | quivers zonotopes CAT zonotopal complexes finite dimensional algebras CAT cube complexes hyperplane arrangements Regular cell complexes left regular bands oriented matroids |
| LCCN | 2022007678 |
| LCCallNum_Ident | QA611.35 .M37 2021 |
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| Notes | November 2021, volume 274, number 1345 (third of 4 numbers) Includes bibliographical references (p. 123-128) and index |
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| Snippet | In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements,
interval greedoids, matroids and... View the abstract. |
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| SubjectTerms | 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 |
| TableOfContents | 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 |
| Title | Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry |
| URI | https://www.ams.org/memo/1345/ https://cir.nii.ac.jp/crid/1130573251094305170 https://ebookcentral.proquest.com/lib/[SITE_ID]/detail.action?docID=6852914 https://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9781470469146 |
| Volume | 274 |
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