On primordial groups for the Green ring

TL;DR

This paper proves that the class of primordial groups for the Green ring of finitely generated kG-modules of trivial source coincides with the family of k-Dress groups, confirming a conjecture for this specific subfunctor.

Contribution

It establishes that the primordial groups for the Green ring of trivial source modules are exactly the k-Dress groups, extending Thevenaz's 1988 prediction.

Findings

Primordial groups for the trivial source Green ring are k-Dress groups.

Confirms Thevenaz's conjecture for a specific subfunctor.

Advances understanding of the structure of Green rings in modular representation theory.

Abstract

Consider the Mackey functor assigning to each finite group G the Green ring of finitely generated kG-modules, where k is a field of characteristic p>0. Thevenaz foresaw in 1988 that the class of primordial groups for this functor is the family of k-Dress groups. In this paper we prove that this is true for the subfunctor defined by the Green ring of finitely generated kG-modules of trivial source.