A new type of diagrams for modules

TL;DR

This paper introduces a novel diagram type called the 'central tuned diagram' for finitely generated modules, utilizing the concept of 'the virtual category,' with applications in modular representations of finite groups of Lie type.

Contribution

It presents the existence proof of the 'central tuned diagram' and introduces the idea of 'the virtual category' for modules, advancing module diagram theory.

Findings

Existence of the 'central tuned diagram' for certain modules.

Introduction of 'the virtual category' concept.

Potential applications in modular representation theory.

Abstract

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.