Isaías David Marín Gaviria

National University of Colombia | UNAL · Escuela de Matemáticas (Medellín)

Currently, researching the Yang–Baxter equation (YBE) is a subject of great interest among scientists of diverse areas in mathematics and other sciences. One of the fundamental open problems is to find all of its solutions. The investigation deals with developing theories such as knot theory, Hopf algebras, quandles, Lie and Jordan (super) algebras...

Representation group theory plays an important role in group theory, much more in geometry, on which we can determine, in many cases, what kind of geometry or geometries a manifold M supports by knowing a faithful representation of the fundamental group Π1(M) in the group Iso + (M) of preserving-orientations isometries of M. Motivated by this and b...

Zavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra A. In this setting, we prove that the indecomposable uniserial A-modules with a mast of multiplicity one...

In this paper, it is proved that the algorithms of differentiation VIII-X (introduced by A.G. Zavadskij to classify equipped posets of tame representation type) induce categorical equivalences between some quotient categories, in particular, an algorithm A z is introduced to build equipped posets with a pair of points (a, b) suitable for differenti...

The enumeration of Dyck paths is one of the most remarkable problems in Catalan combinatorics. Recently introduced categories of Dyck paths have allowed interactions between the theory of representation of algebras and cluster algebras theory. As another application of Dyck paths theory, we present Brauer configurations, whose polygons are defined...

Bijections between invariants associated with indecomposable projective modules over some suitable Brauer configuration algebras and invariants associated with solutions of the Kronecker problem are used to categorify integer sequences in the sense of Ringel and Fahr. Dimensions of the Brauer configuration algebras and their corresponding centers i...

The energy E(G) of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determi...

We introduce an algorithm based on posets and tiled orders to generate emerging images. Experimental results allow concluding that images obtained with these kinds of tools are easy to detect by human beings. It is worth pointing out that the emergence phenomenon is a Gestalt grouping law associated with AI open problems. For this reason, emerging...

Mutations on Brauer configurations are introduced and associated with some suitable automata to solve generalizations of the Chicken McNugget problem. Additionally, based on marked order polytopes, the new Diophantine equations called Gelfand–Tsetlin equations are also solved. The approach allows algebraic descriptions of some properties of the AES...

Let k be a fixed algebraically closed field of arbitrary characteristic, let Λ be a finite dimensional self-injective k-algebra, and let V be an indecomposable non-projective left Λ-module with finite dimension over k. We prove that if τ_Λ V is the Auslander–Reiten translation of V, then the versal deformation rings R(Λ,V) and R(Λ, τ_Λ V ) (in the...

Mutations on Brauer configurations are introduced and associated with some suitable automata in order to solve generalizations of the Chicken McNugget problem. Besides, based on marked order polytopes a new class of diophantine equations called Gelfand-Tsetlin equations are also solved. The approach allows giving an algebraic description of the sch...

Frieze patterns are defined by objects of a category of Dyck paths, to do that, it is introduced the notion of diamond of Dynkin type \({\mathbb {A}}_{n}\). Such diamonds constitute a tool to build integral frieze patterns.

Frieze patterns are defined by objects of a category of Dyck paths, to do that, it is introduced the notion of diamond of Dynkin type An. Such diamonds constitute a tool to build integral frieze patterns.

Bijections between invariants associated to indecomposable projective modules over some suitable Brauer configuration algebras and invariants associated to solutions of the Kronecker problem and the four subspace problem are used to categorify integer sequences in the sense of Ringel and Fahr. Dimensions of the Brauer configuration algebras and the...

In this paper, Krawtchouk-Zavadskij matrices are used to solve systems of differential equations (linear of second order and non-linear), to do that, it is given a matrix interpretation to some identities arising from the classical calculus. We recall that, Krawtchouk-Zavadskij matrices were introduced in the late 1920s by Krawtchouk, however we a...

In this paper, Delannoy numbers are interpreted as dimensions of suitable representations of some equipped posets induced by compositions of integer numbers.

A categorification of the sequence A052558 in the OEIS is given by defining new invariants for indecomposable Kronecker modules.

A complete description of the indecomposable representations and irreducible morphisms of some equipped posets of finite growth representation type is provided.

We prove that the algorithm of differentiation VIII for equipped posets induces a categorical equivalence between quotient categories.

We establish a categorical equivalence induced by the algorithm of differentiation D-IX for equipped posets.

Recently, Vanegas and the first author introduced an algorithm to generate a large amount of emerging images. Such an algorithm uses linear representations of posets and admissible transformations of matrix representations to obtain different kind of gestalts. In this paper, we present an algorithm to extract gestalts of different types from these...