Dynkin Functions and Its Applications
Dynkin functions were introduced by Ringel as a tool to investigate combinatorial properties of hereditary artin algebras. According to Ringel, a Dynkin function consists of four sequences associated to An, Bn, Cn, Dn and five single values associated to the diagrams E6, E7, E8, F4 and G2. He also p...
- Autores:
-
Bravo Rios, Gabriel
- Tipo de recurso:
- Doctoral thesis
- Fecha de publicación:
- 2020
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/79480
- Palabra clave:
- 510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas
Auslander-Reiten quiver
Categorification
Brauer configuration
Brauer configuration algebra
Catalan triangle
Cluster algebras
Dyck paths
Dynkin algebra
Dynkin function
Frieze patterns
Lattice path
Mutation class
Perfect matchings
Poset
Quiver representation
Section
Triangulations
Carcaj de Auslander-Reiten
Configuración de Brauer
Álgebra de Configuración de Brauer
Triángulo de Catalan
Álgebra de Conglomerado
Caminos de Dyck
Álgebra Dynkin
Función Dynkin
Patrones de frizo
Camino reticular
Clases de mutación
Emparejamiento perfecto
Conjunto parcialmente ordenado
Representación de carcaj
Sección
Triangulaciones
Categorización
Análisis matemático
Mathematical analysis
- Rights
- openAccess
- License
- Atribución-NoComercial-SinDerivadas 4.0 Internacional
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Dynkin Functions and Its Applications |
| dc.title.translated.spa.fl_str_mv |
Funciones Dynkin y sus Aplicaciones |
| title |
Dynkin Functions and Its Applications |
| spellingShingle |
Dynkin Functions and Its Applications 510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas Auslander-Reiten quiver Categorification Brauer configuration Brauer configuration algebra Catalan triangle Cluster algebras Dyck paths Dynkin algebra Dynkin function Frieze patterns Lattice path Mutation class Perfect matchings Poset Quiver representation Section Triangulations Carcaj de Auslander-Reiten Configuración de Brauer Álgebra de Configuración de Brauer Triángulo de Catalan Álgebra de Conglomerado Caminos de Dyck Álgebra Dynkin Función Dynkin Patrones de frizo Camino reticular Clases de mutación Emparejamiento perfecto Conjunto parcialmente ordenado Representación de carcaj Sección Triangulaciones Categorización Análisis matemático Mathematical analysis |
| title_short |
Dynkin Functions and Its Applications |
| title_full |
Dynkin Functions and Its Applications |
| title_fullStr |
Dynkin Functions and Its Applications |
| title_full_unstemmed |
Dynkin Functions and Its Applications |
| title_sort |
Dynkin Functions and Its Applications |
| dc.creator.fl_str_mv |
Bravo Rios, Gabriel |
| dc.contributor.advisor.none.fl_str_mv |
Moreno Cañadas, Agustín |
| dc.contributor.author.none.fl_str_mv |
Bravo Rios, Gabriel |
| dc.contributor.researchgroup.spa.fl_str_mv |
TERENUFIA-UNAL |
| dc.subject.ddc.spa.fl_str_mv |
510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas |
| topic |
510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas Auslander-Reiten quiver Categorification Brauer configuration Brauer configuration algebra Catalan triangle Cluster algebras Dyck paths Dynkin algebra Dynkin function Frieze patterns Lattice path Mutation class Perfect matchings Poset Quiver representation Section Triangulations Carcaj de Auslander-Reiten Configuración de Brauer Álgebra de Configuración de Brauer Triángulo de Catalan Álgebra de Conglomerado Caminos de Dyck Álgebra Dynkin Función Dynkin Patrones de frizo Camino reticular Clases de mutación Emparejamiento perfecto Conjunto parcialmente ordenado Representación de carcaj Sección Triangulaciones Categorización Análisis matemático Mathematical analysis |
| dc.subject.proposal.eng.fl_str_mv |
Auslander-Reiten quiver Categorification Brauer configuration Brauer configuration algebra Catalan triangle Cluster algebras Dyck paths Dynkin algebra Dynkin function Frieze patterns Lattice path Mutation class Perfect matchings Poset Quiver representation Section Triangulations |
| dc.subject.proposal.spa.fl_str_mv |
Carcaj de Auslander-Reiten Configuración de Brauer Álgebra de Configuración de Brauer Triángulo de Catalan Álgebra de Conglomerado Caminos de Dyck Álgebra Dynkin Función Dynkin Patrones de frizo Camino reticular Clases de mutación Emparejamiento perfecto Conjunto parcialmente ordenado Representación de carcaj Sección Triangulaciones Categorización |
| dc.subject.unesco.none.fl_str_mv |
Análisis matemático Mathematical analysis |
| description |
Dynkin functions were introduced by Ringel as a tool to investigate combinatorial properties of hereditary artin algebras. According to Ringel, a Dynkin function consists of four sequences associated to An, Bn, Cn, Dn and five single values associated to the diagrams E6, E7, E8, F4 and G2. He also proposes to create an On-line Encyclopedia of Dynkin functions (OEDF) with the same purposes as the famous OEIS. Dynkin functions arise from the context of categorification of integer sequences, which according to Ringel and Fahr it means to consider suitable objects in a category instead of numbers of a given integer sequence. They gave a categorification of Fibonacci numbers by using the Gabriel's universal covering theory and the structure of the Auslander-Reiten quiver of the 3-Kronecker quiver. For instance, if Λ denotes a hereditary artin algebra associated to a Dynkin diagram ∆n then r(∆n) the number of indecomposable modules, a(∆n) the number of antichains in mod Λ, and tn(∆n) the number of tilting modules are Dynkin functions. In particular, we are focused on the way that some Dynkin functions act on Dynkin diagrams of type An. In this work, we follow the ideas of Ringel regarding Dynkin functions by investigating the number of sections in the Auslander-Reiten quiver of algebras of finite representation type. Dyck paths categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin type An and it is shown an algebraic interpretation of frieze patterns as a direct sum of indecomposable objects of the category of Dyck paths. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake graphs. The approach allows us to give formulas for cluster variables in cluster algebras of Dynkin type An in terms of Dyck paths. At last but not least, it is introduced some Brauer configuration algebras such that the dimension of these algebras and its corresponding centers can be obtained via some combinatorial properties of the Catalan triangle. This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 785 de 2017. |
| publishDate |
2020 |
| dc.date.issued.none.fl_str_mv |
2020-10 |
| dc.date.accessioned.none.fl_str_mv |
2021-05-05T18:44:11Z |
| dc.date.available.none.fl_str_mv |
2021-05-05T18:44:11Z |
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Trabajo de grado - Doctorado |
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info:eu-repo/semantics/doctoralThesis |
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https://repositorio.unal.edu.co/handle/unal/79480 |
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Universidad Nacional de Colombia |
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Repositorio Institucional Universidad Nacional de Colombia |
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https://repositorio.unal.edu.co/ |
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Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
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eng |
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eng |
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Atribución-NoComercial-SinDerivadas 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Moreno Cañadas, Agustín9ca55eaf75ecd87559010093e719d1f8Bravo Rios, Gabriel312d39a307e1f8bee7cbdbb979420cfdTERENUFIA-UNAL2021-05-05T18:44:11Z2021-05-05T18:44:11Z2020-10https://repositorio.unal.edu.co/handle/unal/79480Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/Dynkin functions were introduced by Ringel as a tool to investigate combinatorial properties of hereditary artin algebras. According to Ringel, a Dynkin function consists of four sequences associated to An, Bn, Cn, Dn and five single values associated to the diagrams E6, E7, E8, F4 and G2. He also proposes to create an On-line Encyclopedia of Dynkin functions (OEDF) with the same purposes as the famous OEIS. Dynkin functions arise from the context of categorification of integer sequences, which according to Ringel and Fahr it means to consider suitable objects in a category instead of numbers of a given integer sequence. They gave a categorification of Fibonacci numbers by using the Gabriel's universal covering theory and the structure of the Auslander-Reiten quiver of the 3-Kronecker quiver. For instance, if Λ denotes a hereditary artin algebra associated to a Dynkin diagram ∆n then r(∆n) the number of indecomposable modules, a(∆n) the number of antichains in mod Λ, and tn(∆n) the number of tilting modules are Dynkin functions. In particular, we are focused on the way that some Dynkin functions act on Dynkin diagrams of type An. In this work, we follow the ideas of Ringel regarding Dynkin functions by investigating the number of sections in the Auslander-Reiten quiver of algebras of finite representation type. Dyck paths categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin type An and it is shown an algebraic interpretation of frieze patterns as a direct sum of indecomposable objects of the category of Dyck paths. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake graphs. The approach allows us to give formulas for cluster variables in cluster algebras of Dynkin type An in terms of Dyck paths. At last but not least, it is introduced some Brauer configuration algebras such that the dimension of these algebras and its corresponding centers can be obtained via some combinatorial properties of the Catalan triangle. This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 785 de 2017.Las funciones Dynkin fueron introducidas por Ringel como una herramienta para investigar las propiedades combinatorias de las álgebras hereditarias de artin. Según Ringel, una función Dynkin consta de cuatro sucesiones asociadas a An, Bn, Cn, Dn y cinco valores únicos asociados a los diagramas E6, E7, E8, F4 y G2. También propone crear una Enciclopedia en línea de funciones Dynkin (OEDF) con los mismos propósitos que la famosa OEIS. Las funciones Dynkin surgen del contexto de categorización de sucesiones enteras, que según Ringel y Fahr significa considerar objetos adecuados en una categoría en lugar de números de una sucesión entera dada. Ellos dieron una categorización de los números de Fibonacci utilizando la teoría de cubrimiento universal de Gabriel y la estructura del carcaj Auslander-Reiten del carcaj 3-Kronecker. Por ejemplo, si Λ denota una álgebra hereditaria de artin asociada a un diagrama de Dynkin ∆n entonces r (∆n) el número de módulos indescomponibles, a(∆n) el número de anticadenas en mod Λ, y tn (∆n) el número de módulos inclinantes son funciones Dynkin. En particular, nos centramos en la forma en que algunas funciones Dynkin actúan en los diagramas de Dynkin de tipo An. En este trabajo, seguimos las ideas de Ringel con respecto a las funciones Dynkin investigando el número de secciones en el carcaj de Auslander-Reiten de álgebras de tipo representación finita. Las categorías de caminos de Dyck se introducen como un modelo combinatorio de la categoría de representaciones de carcajes de tipo Dynkin An y se muestra una interpretación algebraica de patrones de friso como una suma directa de objetos indescomponibles de la categoría de caminos de Dyck. En particular, se ha demostrado que existe una biyección entre algunas caminos de Dyck y emparejamientos perfectos de algunos grafos serpientes. El enfoque nos permite dar fórmulas para las variables de conglomerado en álgebras de conglomerado Dynkin de tipo An en términos de caminos de Dyck. Por último, pero no menos importante, se introducen algunas álgebras de configuración de Brauer de modo que la dimensión de estas álgebras y sus correspondientes centros se puede obtener mediante algunas propiedades combinatorias del triángulo de catalán. Esta investigación fue apoyada parcialmente por COLCIENCIAS convocatoria doctorados nacionales 785 de 2017.DoctoradoTeoría de representaciones de álgebras1 recurso en línea (135 páginas)application/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Doctorado en Ciencias - MatemáticasDepartamento de MatemáticasFacultad de CienciasBogotáUniversidad Nacional de Colombia - Sede Bogotá510 - Matemáticas::519 - Probabilidades y matemáticas aplicadasAuslander-Reiten quiverCategorificationBrauer configurationBrauer configuration algebraCatalan triangleCluster algebrasDyck pathsDynkin algebraDynkin functionFrieze patternsLattice pathMutation classPerfect matchingsPosetQuiver representationSectionTriangulationsCarcaj de Auslander-ReitenConfiguración de BrauerÁlgebra de Configuración de BrauerTriángulo de CatalanÁlgebra de ConglomeradoCaminos de DyckÁlgebra DynkinFunción DynkinPatrones de frizoCamino reticularClases de mutaciónEmparejamiento perfectoConjunto parcialmente ordenadoRepresentación de carcajSecciónTriangulacionesCategorizaciónAnálisis matemáticoMathematical analysisDynkin Functions and Its ApplicationsFunciones Dynkin y sus AplicacionesTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttp://purl.org/redcol/resource_type/TD[1] G. 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Ser. 376 (2005), 413-436.Dynkin Functions and Its ApplicationsCOLCIENCIASORIGINAL1012346226.2020.pdf1012346226.2020.pdfTesis de Doctorado en Ciencias - Matemáticasapplication/pdf1031279https://repositorio.unal.edu.co/bitstream/unal/79480/1/1012346226.2020.pdf2c1f2dd4ffc33d13fb7559230a8566dbMD51LICENSElicense.txtlicense.txttext/plain; charset=utf-83964https://repositorio.unal.edu.co/bitstream/unal/79480/2/license.txtcccfe52f796b7c63423298c2d3365fc6MD52CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8805https://repositorio.unal.edu.co/bitstream/unal/79480/3/license_rdf4460e5956bc1d1639be9ae6146a50347MD53THUMBNAIL1012346226.2020.pdf.jpg1012346226.2020.pdf.jpgGenerated Thumbnailimage/jpeg3759https://repositorio.unal.edu.co/bitstream/unal/79480/4/1012346226.2020.pdf.jpge7e278831d6705906ceed4c3ab002b27MD54unal/79480oai:repositorio.unal.edu.co:unal/794802023-07-29 23:03:54.222Repositorio Institucional Universidad Nacional de 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