arXiv subjects

Stephanos Gekas

Publications and source records attributed to Stephanos Gekas.

3 recordsLinked to original sources

arXiv2017

From Subdirect Sums to Virtuality

We start by an original investigation on subgroups of (even infinite) direct sums in the first 4 sections, that largely generalizes Remak's known theorem; inspired by that general picture we have elsewhere extended this elementary "virtual" diagrammatic situation (in diagrammatic length 2 meaning set-theoretic fixation of vertices) by generalizing to the notion of "virtuality" in module extensions and diagrams in modular representation theory. Our first approach starts with an appropriately defined equivalence relation, which is precisely what allows for treating the confusing case of multiple factors, thus giving a deeper insight into the structure of such subgroups. Several applications and new techniques arising from that approach are examined, even ones concerning basic properties of homomorphisms, extending well-known elementary ones.

math.GR↗

arXiv2017

Virtual Extensions of Modules

In this article we are examining extensions and some basic diagrammatic properties of modules, in both cases from a new, "virtual" point of view. As natural background for investigating the kind of problems we are dealing with, the virtual category of a module M is introduced, having as objects the submodules of M's subquotients modulo some identifications. In the case of extensions our approach implies viewing "proportionality classes" of extensions of (dually, by) a simple module by (resp. of) another simple as quotients of a certain quotient (which is in fact a subdirect product) of a projective cover, that comprises all those classes - or dually as submodules of a comprising submodule (which is a push-out) of an injective hull. In particular we become thus able to upgrade the Yoneda correspondence to a bimodule isomorphism. Basic steps toward the foundation of and investigation into the theory of Virtual Diagrams are also made here. In particular, the "virtuality group" A(D) of a virtual diagram D of a module M is introduced, generated by the D-visible virtual constituents of M, with respect to an addition that generalizes the one of submodules in a module.

math.RT↗

arXiv2016

A new type of diagrams for modules

We introduce a new type of diagrams and prove the existence of a particular one, the "central tuned diagram", with some optimal features, for finitely generated modules of certain categories. This is achieved by getting to the idea of "the virtual category" of a module. Important applications are specifically suggested to the modular representations of finite groups of Lie type.

math.RT↗