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...
Uložené v:
| Hlavní autori: | , , |
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| Médium: | E-kniha Kniha |
| Jazyk: | English |
| Vydavateľské údaje: |
Providence, Rhode Island
American Mathematical Society
2022
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| 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
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| ISBN: | 9781470450427, 1470450429 |
| ISSN: | 0065-9266, 1947-6221 |
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| Shrnutí: | 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. |
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| Bibliografia: | November 2021, volume 274, number 1345 (third of 4 numbers) Includes bibliographical references (p. 123-128) and index |
| ISBN: | 9781470450427 1470450429 |
| ISSN: | 0065-9266 1947-6221 |
| DOI: | 10.1090/memo/1345 |