Sym(n)- and Alt(n)-modules with an additive dimension

TL;DR

This paper generalizes classical results on minimal symmetric and alternating group modules by introducing a new notion of modules with an additive dimension, applicable across various mathematical frameworks.

Contribution

It presents a unified approach to classify minimal Sym(n)- and Alt(n)-modules using the concept of modules with an additive dimension, extending classical and modern settings.

Findings

Fully classifies faithful Sym(n)- and Alt(n)-modules of least dimension

Introduces a new notion of modules with an additive dimension

Bridges classical, o-minimal, and finite Morley rank contexts

Abstract

We revisit, clarify, and generalise classical results of Dickson and (much later) Wagner on minimal Sym(n)- and Alt(n)-modules. We present a new, natural notion of 'modules with an additive dimension' covering at once the classical, finitary case as well as modules definable in an o-minimal or finite Morley rank setting; in this context, we fully identify the faithful Sym(n)- and Alt(n)-modules of least dimension.