The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications Isa´ıas David Mar´ın Gaviria Universidad Nacional de Colombia Facultad de Ciencias Departamento de Matematicas´ Bogota,´ D.C. November, 2019 The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications Isa´ıas David Mar´ın Gaviria Thesis work to obtain the degree of Doctor in Mathematics Advisor Professor Octavio Mendoza Hernandez,´ Ph.D. Full professor, National Autonomous University of Mexico-D.F Coadvisor Agust´ın Moreno Canadas,~ Ph.D. Associate Professor, National University of Colombia Research line Representation theory of posets Research group TERENUFIA-UNAL Universidad Nacional de Colombia Facultad de Ciencias Departamento de Matematicas´ Bogota,´ D.C. November, 2019 5 Title The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications Abstract: The theory of representation of partially ordered sets or posets was introduced in the early 1970's as an effort to give an answer to the second Brauer-Thrall conjecture. Recall that one of the main goals of this theory is to give a complete description of the indecomposable objects of the category of representations of a given poset. Perhaps the most useful tool to obtain such classification are the algorithms of differentiation. For instance, Nazarova and Roiter introduced an algorithm known as the algorithm of differentiation with respect to a maximal point which allowed to Kleiner in 1972 to obtain a classification of posets of finite representation type. Soon afterwards between 1974 and 1977, Zavadskij defined the more general algorithm I with respect to a suitable pair of points, this algorithm was used in 1981 by Nazarova and Zavadskij in order to give a criterion for the classification of posets of finite growth representation type. Actually, several years later, Zavadskij himself described the structure of the Auslander-Reiten quiver of this kind of posets, to do that, it was established that such an algorithm is in fact a categorical equivalence. Since the theory of representation of posets was developed in the 1980's and 1990's for posets with additional structures, for example, for posets with involution or for equipped posets by Bondarenko, Nazarova, Roiter, Zabarilo and Zavadskij among others. It was necessary to define a new class of algorithms to classify posets with these additional structures. In fact, Zavadskij introduced 17 algorithms. Algorithms, I-V (and some additional differentiations) were used by him and Bondarenko to classify posets with involution, whereas algorithms I, VII-XVII were used to classify equipped posets. In particular, algorithms I, VII, VIII and IX were used to classify equipped posets of finite growth representation type without paying attention to the behavior of the morphisms of the corresponding categories. In other words, it was obtained a classification of the objects without proving that the algorithms used to tackle the problems are in fact categorical equivalences, therefore, the main problem of the theory of the algorithms of differentiation consists of giving a detailed description of the behavior of the morphisms under these additive functors, such description allows to give a deep understanding of the Auslander-Reiten quiver of the corresponding categories. On the other hand, in the last few years has been noted a great interest in the application of the theory of representation of algebras in different fields of computer science, for example, in combinatorics, information security and topological data analysis. Ringel and Fahr, for instance, gave a categorification of Fibonacci numbers by using the 3-Kronecker quiver whereas representation of posets and the theory of posets have been used to analyze tactics of war and cyberwar. Besides, the theory of Auslander has been used to analyze big data via the homological persistent theory. In this research, it is proved that the algorithms of differentiation VIII-X induce categorical equivalences between some quotient categories, giving a description of the Auslander-Reiten quiver of some equipped posets by using the evolvent associated to these kind of posets. In this work, ideas arising from the theory of representation of equipped posets are used to give a categorification of Delannoy numbers. Actually, such numbers are interpreted as dimensions of some suitable equipped posets. We also interpret the algorithm of differentiation VII as a steganographic algorithm which allows to generate digital watermarks, such an algorithm can be also used to describe the behavior of some kind of informatics viruses, in fact, it is explained how this algorithm describe the infection-detection process when a computer network is affected for this type of malware. At last but not least, we recall that the theory of representation of equipped posets is a way to deal with the homogeneous biquadratic problem which is an open matrix problem, in this case, with respect to a pair of fields (F; G) with G a quadratic extension of the field F with respect to a polynomial of the form t2 + q, q 2 F . Actually, explicit solutions to this problem were given by Zavadskij who rediscovered in 2007 the Krawtchouk matrices introducing an interesting Θ-transformation as well. In this research, such Krawtchouk matrices are used in order to give explicit solutions to non-linear systems of differential equations of the form X0(t) + AX2(t) = B, where X(t);X0(t);A and B are n × n square matrices. Tools arising from this solution are called in this work the Zavadskij calculus. This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 727 de 2015. Keywords: Representation theory of partially ordered sets, Representation theory of quivers, Auslander-Reiten theory, Matrix problem, Equipped poset, Equipped posets with Involution, Algorithms of Differentiation, Delannoy numbers, Zavadskij Calculus, Partitions of integer numbers, Integer sequences, Digital Watermark, Virus and Worms. Acceptation Note Thesis Work \ mention" Jury Jury Jury Advisor Octavio Mendoza Hern´andez Coadvisor Agust´ınMoreno Ca~nadas Bogot´a,D.C., November, 2019 Dedicated to My mom Luz Dary Gaviria and to my wife Andrea Valencia for their constant support and who never gave up on me. A.M. Ca~nadas, who put me on the road, who accompanied me throughout the journey and encouraged me in difficult times. Acknowledgments I would like to express my special appreciation to my advisor Professor Agustin Moreno, Ph.D. for being such a tremendous mentor for me. I would like to thank you for encourag- ing my research and for allowing me to grow as a research and as a scientist. His advice on both research as well as on my career have been invaluable. I would also like to thank my other advisor, Professor Octavio Mendoza, Ph.D. for his hard work and thorough review of this thesis. I also want to thank him for his brilliant comments and suggestions, so that this work could be read in a clearer and more scientific way, thanks for his support and discussion sessions during my visit research internship in National Autonomous University of Mexico in 2017, many thanks. A special thanks to my family, words cannot express how grateful I am to my mother Luz Dary, for all of the sacrifices she has made for my well-being on my behalf. Her prayer for me was what sustained me thus far. I would also like to thank my beloved wife Andrea Valencia, MSc. Thanks for always supporting me, for allowing me to sacrifice our family time to finish this research, she took care of me in times where I used to work too much. I specially express my regards to my partners of the TERENUFIA-UNAL research group for the stimulating discussions, for their feedback, for their contributions and mainly for encouraging me throughout this experience. I also want to express my gratitude to my close friends PFF Espinosa and VC Vargas for being such good friends, for always motivating me and for being with me throughout this process, also for providing discussion spaces for the advance of this investigation. Last but not least, I would like to thank to Professor A.G. Zavadskij, who inspired all this research by leaving us his legacy of research. Finally, I thank God, for letting me to overcome all the difficulties. I have experienced His guidance day by day; He was the one who let me finish my doctorate studies. Thanks My Lord. Contents Contents III List of Tables VII List of Figures IX Introduction XIII 1. Partially ordered sets with additional structure 1 1.1 Equipped posets and equipped posets with involution . .1 1.2 Categories . .5 1.2.1 Auslander-Reiten theory in a Krull-Schmidth category . .9 1.3 Complexification and reellification . 12 1.4 Representation of equipped posets and equipped posets with involution . 15 1.5 The matrix problem . 17 1.6 Some indecomposable objects . 19 1.7 Some classification theorems . 22 2. Algorithms of differentiation 27 2.1 (A; B)-cleaving pairs and the Zavadskij symbol . 27 2.2 Algorithm of differentiation I with respect to a suitable pair of points and the completion algorithm . 28 2.2.1 Algorithm of differentiation I . 29 2.2.2 Completion algorithm . 30 2.3 Categorical properties of differentiation VII for equipped posets . 32 2.4 Categorical properties of the algorithm of differentiation VIII for equipped posets.................................................. 37 2.5 Categorical properties of the algorithm of differentiation IX for equipped posets.................................................. 44 III CONTENTS IV 2.6 Categorical properties of the algorithm of differentiation X for equipped posets with involution . 50 3. The Auslander-Reiten quiver of some equipped posets 58 3.1 The lifting algorithm . 58 3.2 The Auslander-Reiten quiver of the equipped posets F13 − F18 .........