Abstract 3D-rotation groups and recognition of icosahedral modules

TL;DR

This paper introduces a novel abstract concept of 3D-rotation modules for groups, characterizes the finite groups with such modules, and applies these results to classify certain modules of the alternating group Alt(5).

Contribution

It defines an abstract notion of 3D-rotation modules without vector space assumptions and classifies groups and modules satisfying this concept, extending classical representation theory.

Findings

Only known finite groups (Alt(4), Sym(4), Alt(5)) admit such modules.

The module structure for Alt(5) is fully determined under certain conditions.

Classifies Alt(5)-modules with additive dimension, dimension 3, and no 2-torsion.

Abstract

We introduce an abstract notion of a 3D-rotation module for a group $G$ that does not require the module to carry a vector space structure, a priori nor a posteriori. We prove that, under an expected irreducibility-like assumption, the only finite $G$ with such a module are those already known from the classical setting: $\operatorname{Alt}(4)$, $\operatorname{Sym}(4)$, and $\operatorname{Alt}(5)$. Our main result then studies the module structure when $G = \operatorname{Alt}(5)$ and shows that, under certain natural restrictions, it is fully determined and generalizes that of the classical icosahedral module. We include an application to the recently introduced setting of modules with an additive dimension, a general setting allowing for simultaneous treatment of classical representation theory of finite groups as well as representations within various well-behaved model-theoretic settings such as the $o$-minimal and finite Morley rank ones. Leveraging our recognition result for icosahedral modules, we classify the faithful $\operatorname{Alt}(5)$-modules with additive dimension that are dim-connected of dimension $3$ and without $2$-torsion.