Emparejamientos perfectos, álgebras de conglomerado y algunas de sus aplicaciones
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Melo Lopez, Astrid Carolina
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The main objective of this work is to study cluster algebras, perfect matchings of a class of particular graphs, and the various relationships that exist between these topics. In order to do this, concepts and results related to the theory of graphs and the theory of representation of quivers are introduced, which are fundamental to carry out a combinatorial and algebraic study of the subject. Subsequently, the concept of cluster algebras is introduced, and some of its applications are presented, such as the construction of the Auslander-Reiten quiver, sequences and diophantine equations, and counting of perfect matchings through continuous fractions in snake graphs. As a result of the previously mentioned study, the solution of a diophantine equation, the counting of matchings in a family of graphs, and the relationship between matchings, Dyck paths, partitions, triangulations of regular polygons and Aztec diamonds are presented. Finally, some conclusions and recommendations are presented that will serve as a basis for defining future research work.
El objetivo principal de este trabajo es estudiar las álgebras de conglomerado, los emparejamientos perfectos de una clase de grafos particulares, y las diversas relaciones que existen entre estos dos temas. Para esto, se introducen conceptos y resultados relacionados con la teoría de grafos y la teoría de representación de carcajes, los cuales son fundamentales para realizar un estudio combinatorio y algebraico del tema. Posteriormente, se introduce el concepto de álgebras de conglomerado, y se presentan algunas de sus aplicaciones, como lo son la construcción del carcaj de Auslander-Reiten, sucesiones y ecuaciones diofánticas, y conteo de emparejamientos perfectos por medio de fracciones continuas en grafos serpiente. Como resultado del estudio previamente mencionado, se presenta la solución de una ecuación diofántica, el conteo de emparejamientos en una familia de grafos, y la relación entre emparejamientos, caminos de Dyck, particiones, triangulaciones de polígonos regulares y diamantes Aztecas. Por último, se presentan algunas conclusiones y recomendaciones que servirán de base para definir un futuro trabajo de investigación.
El objetivo principal de este trabajo es estudiar las álgebras de conglomerado, los emparejamientos perfectos de una clase de grafos particulares, y las diversas relaciones que existen entre estos dos temas. Para esto, se introducen conceptos y resultados relacionados con la teoría de grafos y la teoría de representación de carcajes, los cuales son fundamentales para realizar un estudio combinatorio y algebraico del tema. Posteriormente, se introduce el concepto de álgebras de conglomerado, y se presentan algunas de sus aplicaciones, como lo son la construcción del carcaj de Auslander-Reiten, sucesiones y ecuaciones diofánticas, y conteo de emparejamientos perfectos por medio de fracciones continuas en grafos serpiente. Como resultado del estudio previamente mencionado, se presenta la solución de una ecuación diofántica, el conteo de emparejamientos en una familia de grafos, y la relación entre emparejamientos, caminos de Dyck, particiones, triangulaciones de polígonos regulares y diamantes Aztecas. Por último, se presentan algunas conclusiones y recomendaciones que servirán de base para definir un futuro trabajo de investigación.
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Emparejamiento perfecto, álgebra de conglomerado, carcaj de Auslander-Reiten, triangulación, camino de Dyck, grafo serpiente, fracción continua, ecuación diofántica; Perfect matching, cluster algebra, Auslander-Reiten's quiver, triangulation, Dyck path, snake graph, continued fraction, diophantic equation
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@BOOK{Assem, author={Assem, Ibrahim and Simson, Daniel and Skowrónski, Andrzej}, title={Elements of the Representation Theory of Associative Algebras. Vol. 1: Techniques of Representation Theory}, edition={}, publisher={London Mathematical Society\linebreak Student Texts}, address={Cambridge University Press, Cambridge}, year={2006}, month = {02}, pages = {}, }
@BOOK{AssemT, author = {Assem, Ibrahim and Trepode, Sonia}, title = {Homological Methods, Representation Theory, and Cluster Algebras}, publisher = {Springer International Publishing}, edition={}, address = {}, year = {2018}, month = {04}, pages = {}, }
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@article{Fomin1, author={Fomin, Sergey and Shapiro, Michael and Thurston, Dylan}, year = {2006}, month = {08}, pages = {}, title={Cluster algebras and triangulated surfaces. Part I: Cluster complexes}, volume = {}, journal = {arXiv Mathematics e-prints}, doi = {}, }
@article{Fomin2, author = {Fomin, Sergey and Zelevinsky, Andrei}, year = {2001}, month = {05}, pages = {}, title = {Cluster algebras I: Foundations}, volume = {15}, journal = {Journal of the American Mathematical Society}, doi = {10.1090/S0894-0347-01-00385-X} }
@article{Fomin3, author = {Fomin, Sergey and Zelevinsky, Andrei}, year = {2003}, month = {01}, pages = {}, title = {Cluster algebras II: Finite type classification}, volume = {154}, journal = {Inventiones\linebreak mathematicae}, doi = {10.1007/s00222-003-0302-y} }
@article{Fomin4, author={Fomin, Sergey and Williams, Lauren and Zelevinsky, Andrei}, year = {2017}, month = {}, pages = {}, title={Introduction to Cluster Algebras. Chapters 1-3}, volume = {}, journal = {arXiv Mathematics e-prints}, doi = {}, }
@article{fomin2005root, title={Root systems and generalized\linebreak associahedra}, author={Sergey Fomin and Nathan Reading}, year={2005}, month = {10}, journal = {arXiv Mathematics e-prints}, eprint={math/0505518}, archivePrefix={arXiv}, primaryClass={math.CO} }
@article{Gessel, author = {Gessel, Ira and Viennot, Xavier}, year = {2000}, month = {09}, pages = {}, title = {Determinants, Paths, and Plane Partitions}, journal = {De Mathematiques, Departement} }
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@article{Keller, author={Keller, Bernhard}, year = {2008}, month = {07}, pages = {}, title={Cluster Algebras, Quiver Representations and Triangulated \linebreak Categories}, journal = {arXiv Mathematics e-prints}, doi = {arXiv:0807.1960}, }
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@BOOK{Lampe, author={Lampe, Philipp}, title={Cluster Algebras}, publisher={Department of Mathematical Sciences, Durham University}, edition={}, address={United Kingdom}, year={2013} }
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@BOOK{Rotman, author={Rotman, Joseph}, title={An Introduction to Homological Algebra}, publisher={\linebreak Springer International Publishing}, address={Academic Press}, edition={}, year={2009} }
@BOOK{Schiffler1, author={Schiffler, Ralf}, title={Quiver Representations}, publisher={Canadian Mathematical Society-Department of Mathematics University of Connecticut, CMS Books in\linebreak Mathematics, Springer International Publishing}, edition={}, address={}, year={2014} }
@article{Schiffler2, author={Schiffler, Ralf}, year={2008}, month = {08}, pages = {}, title={A Geometric Model for Cluster Categories of Type $\mathbb{D}_{n}$}, volume = {}, journal = {arXiv Mathematics e-prints}, doi = {} }
@article{Schiffler3, author={Çanakçi, Ilke and Schiffler, Ralf}, year={2016}, month = {08}, pages = {}, title={Cluster Algebras and Continued Fractions}, volume = {}, journal = {arXiv e-prints}, doi = {10.1112/S0010437X17007631} }
@article{Schiffler4, author={Çanakçi, Ilke and Schiffler, Ralf}, year = {2012}, month = {09}, pages = {}, title={Snake graph calculus and cluster algebras from surfaces}, volume = {154}, journal = {arXiv e-prints}, doi = {} }
@article{Schiffler5, author={Caldero, Philippe and Chapoton, Frederic and Schiffler, Ralf}, year = {2006}, month = {01}, pages = {}, title={Quivers with relations arising from clusters ($\mathbb{A}_n$ case)}, volume = {154}, journal = {Transactions of the A.M.S.}, doi = {10.1090/S0002-9947-05-03753-0} }
@book{Stanley0, author={Stanley, Richard}, title={Catalan Numbers}, publisher={Cambridge University Press}, place={Cambridge}, DOI={10.1017/CBO9781139871495}, year={2015} }
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@BOOK{Vermani, author={Vermani, L.R.}, title={An Elementary Approach to Homological Algebra}, publisher={Chapman $\&$ Hall/CRC, Monographs and Surveys in Pure and Applied Mathematics}, address={}, edition={}, year={2003} }
@BOOK{Assem, author={Assem, Ibrahim and Simson, Daniel and Skowrónski, Andrzej}, title={Elements of the Representation Theory of Associative Algebras. Vol. 1: Techniques of Representation Theory}, edition={}, publisher={London Mathematical Society\linebreak Student Texts}, address={Cambridge University Press, Cambridge}, year={2006}, month = {02}, pages = {}, }
@BOOK{AssemT, author = {Assem, Ibrahim and Trepode, Sonia}, title = {Homological Methods, Representation Theory, and Cluster Algebras}, publisher = {Springer International Publishing}, edition={}, address = {}, year = {2018}, month = {04}, pages = {}, }
@ARTICLE{aztec, author = {Elkies, Noam and Kuperberg, Greg and Larsen, Michael and Propp, James}, year = {1991}, month = {06}, pages = {}, title = {Alternating sign matrices and domino tilings}, volume = {}, journal = {arXiv Mathematics e-prints} }
@ARTICLE{Eu, author = {Eu, Sen-Peng and Fu, Tung-Shan}, year = {2005}, month = {01}, pages = {}, title = {A Simple Proof of the Aztec Diamond Theorem}, volume = {12}, journal = {Electronic Journal of Combinatorics} }
@article{Fomin1, author={Fomin, Sergey and Shapiro, Michael and Thurston, Dylan}, year = {2006}, month = {08}, pages = {}, title={Cluster algebras and triangulated surfaces. Part I: Cluster complexes}, volume = {}, journal = {arXiv Mathematics e-prints}, doi = {}, }
@article{Fomin2, author = {Fomin, Sergey and Zelevinsky, Andrei}, year = {2001}, month = {05}, pages = {}, title = {Cluster algebras I: Foundations}, volume = {15}, journal = {Journal of the American Mathematical Society}, doi = {10.1090/S0894-0347-01-00385-X} }
@article{Fomin3, author = {Fomin, Sergey and Zelevinsky, Andrei}, year = {2003}, month = {01}, pages = {}, title = {Cluster algebras II: Finite type classification}, volume = {154}, journal = {Inventiones\linebreak mathematicae}, doi = {10.1007/s00222-003-0302-y} }
@article{Fomin4, author={Fomin, Sergey and Williams, Lauren and Zelevinsky, Andrei}, year = {2017}, month = {}, pages = {}, title={Introduction to Cluster Algebras. Chapters 1-3}, volume = {}, journal = {arXiv Mathematics e-prints}, doi = {}, }
@article{fomin2005root, title={Root systems and generalized\linebreak associahedra}, author={Sergey Fomin and Nathan Reading}, year={2005}, month = {10}, journal = {arXiv Mathematics e-prints}, eprint={math/0505518}, archivePrefix={arXiv}, primaryClass={math.CO} }
@article{Gessel, author = {Gessel, Ira and Viennot, Xavier}, year = {2000}, month = {09}, pages = {}, title = {Determinants, Paths, and Plane Partitions}, journal = {De Mathematiques, Departement} }
@article{Kasteleyn, author={P. W. Kasteleyn}, year = {1961}, month = {}, pages = {1209–1225}, title={The statistics of dimers on a lattice I. The number of dimer arrangements on a quadratic lattice}, volume = {27}, journal = {Physica}, doi = {}, }
@article{Keller, author={Keller, Bernhard}, year = {2008}, month = {07}, pages = {}, title={Cluster Algebras, Quiver Representations and Triangulated \linebreak Categories}, journal = {arXiv Mathematics e-prints}, doi = {arXiv:0807.1960}, }
@BOOK{Koshy, author={Koshy, Thomas}, title={Fibonacci and Lucas Numbers with Applications}, publisher={Pure and applied mathematics: a Wiley-Interscience series of texts, monographs, and tract}, edition={}, address={New York}, year={2001}, }
@BOOK{Lampe, author={Lampe, Philipp}, title={Cluster Algebras}, publisher={Department of Mathematical Sciences, Durham University}, edition={}, address={United Kingdom}, year={2013} }
@BOOK{Matching, author={Lovász, László and Plummer, M. D.}, title={Matching Theory}, publisher={North-Holland Mathematics Studies}, address={Rutgers University, New Brunswick, NJ, U.S.A}, edition={}, year={1986}, }
@BOOK{Rotman, author={Rotman, Joseph}, title={An Introduction to Homological Algebra}, publisher={\linebreak Springer International Publishing}, address={Academic Press}, edition={}, year={2009} }
@BOOK{Schiffler1, author={Schiffler, Ralf}, title={Quiver Representations}, publisher={Canadian Mathematical Society-Department of Mathematics University of Connecticut, CMS Books in\linebreak Mathematics, Springer International Publishing}, edition={}, address={}, year={2014} }
@article{Schiffler2, author={Schiffler, Ralf}, year={2008}, month = {08}, pages = {}, title={A Geometric Model for Cluster Categories of Type $\mathbb{D}_{n}$}, volume = {}, journal = {arXiv Mathematics e-prints}, doi = {} }
@article{Schiffler3, author={Çanakçi, Ilke and Schiffler, Ralf}, year={2016}, month = {08}, pages = {}, title={Cluster Algebras and Continued Fractions}, volume = {}, journal = {arXiv e-prints}, doi = {10.1112/S0010437X17007631} }
@article{Schiffler4, author={Çanakçi, Ilke and Schiffler, Ralf}, year = {2012}, month = {09}, pages = {}, title={Snake graph calculus and cluster algebras from surfaces}, volume = {154}, journal = {arXiv e-prints}, doi = {} }
@article{Schiffler5, author={Caldero, Philippe and Chapoton, Frederic and Schiffler, Ralf}, year = {2006}, month = {01}, pages = {}, title={Quivers with relations arising from clusters ($\mathbb{A}_n$ case)}, volume = {154}, journal = {Transactions of the A.M.S.}, doi = {10.1090/S0002-9947-05-03753-0} }
@book{Stanley0, author={Stanley, Richard}, title={Catalan Numbers}, publisher={Cambridge University Press}, place={Cambridge}, DOI={10.1017/CBO9781139871495}, year={2015} }
@BOOK{Stanley1, author={Stanley, Richard}, title={Enumerative Combinatorics, Volume 1}, publisher={Cambridge}, edition={}, address={Cambridge Studies in\linebreak Advanced Mathematics, vol. 49, Cambridge University Press}, year={2012} }
@BOOK{Stanley2, author={Stanley, Richard}, title={Enumerative Combinatorics, Volume 2}, publisher={Cambridge}, edition={}, address={Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press}, year={1999} }
@BOOK{Vermani, author={Vermani, L.R.}, title={An Elementary Approach to Homological Algebra}, publisher={Chapman $\&$ Hall/CRC, Monographs and Surveys in Pure and Applied Mathematics}, address={}, edition={}, year={2003} }