Stable representations of posets and their applications

• Grant number: 15/00116-4
• Support Opportunities: Regular Research Grants
• Start date: July 01, 2015
• End date: June 30, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Kostiantyn Iusenko
• Grantee: Kostiantyn Iusenko
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

In the last forty years representations of finite-dimensional algebras became to a huge field of research. One of the fundamental task in this theory is to describe all finite-dimensional representations (up to isomorphism) of a given algebra. A possible way to do this is to solve the corresponding matrix problem: to put a partitioned matrix into a canonical form using not all elementary operations, but a subset defined by the partition. In such a way L.A.Nazarova and A.V.Roiter defined the representations of posets. Such concept together with developed techniques provide the powerful tools for determining the representation type of algebras and forstudying their indecomposable representations. Nevertheless the full classification of indecomposable representations is rarely possible in the sense that "most" algebras are wild (that is, the problem of classification of their representations is of the same difficulty as the classification of representations offree algebras). One of the possible way to organize the representation theory of wild algebras is via geometrical approach to classification problems. A.King constructed moduli spaces of representation of finite-dimensional algebras using Mumford's geometric invariant theory. In the connection to such approach it is natural to study so-called stable representations of quivers that satisfy certain relations and, in particular, stable representations of posets. Our main goal is to continue the investigation of stable representations of posets. The investigation will be divided into two parts. The first one is representation-theoretical questions: we will investigate the behavior stability of a given poset, stability under Coxeter and differentiation functors, and others questions. The second part is dedicated to applications: we will study certain properties of moduli spaces of stable sheaves over toric varieties using stable representations of posets. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

FUTORNY, VYACHESLAV; IUSENKO, KOSTIANTYN. Stable representations of posets. Journal of Pure and Applied Algebra, v. 223, n. 12, p. 5251-5278, DEC 2019. (14/09310-5, 15/00116-4)

FONSECA, CLAUDIA CAVALCANTE; IUSENKO, KOSTIANTYN. On dimension of poset variety. Linear Algebra and its Applications, v. 568, p. 155-164, MAY 1 2019. (14/09310-5, 15/00116-4)