A Classification of the Isomorphism Types of Indecomposable and Simple Modules that Refines the Green Theory in Finite Group Modular Representation Theory

TL;DR

This paper refines Green Theory by providing a detailed classification of indecomposable and simple modules in finite group modular representation theory, introducing new invariants and formulas for module counts.

Contribution

It introduces a refined classification scheme for modules that enhances Green Theory and derives a new formula for counting absolutely simple modules.

Findings

Modules are classified into disjoint subsets sharing invariants.

A new formula for counting absolutely simple modules is established.

The classification provides deeper insight into module isomorphism types.

Abstract

In Finite Group Modular Representation Theory, the basic objects are the indecomposable and simple modules. This paper offers a new classification of these objects that refines the Green Theory Classification of indecomposable and simple modules. The sets of isomorphism tyes of these modules is decomposed into disjoint, non-empty subsets such that any two elements in any subset share Green Theory Invariants. We also prove a new formula for the number of isomorphism types of absolutely simple modules of finite groups in prime characteristic.