The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications
- Autores: Isaías David Marín Gaviria
- Directores de la Tesis: Agustín Moreno Cañadas (dir. tes.), Octavio Mendoza Hernández (dir. tes.)
- Lectura: En la Universidad Nacional de Colombia (UNAL) ( Colombia ) en 2020
- Idioma: español
- Materias:
- Enlaces
- Tesis en acceso abierto en: repositorio.unal.edu.co
- Resumen
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, 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 t^2 + q, q∈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 X'(t)+AX^2 (t)=B, where X(t); X'(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.
