Positive representations of finite groups in Riesz spaces

TL;DR

This paper develops the theory of positive representations of finite groups in Riesz spaces, characterizing their structure, irreducibility, and how they differ from classical representation theory, including the breakdown of character theory.

Contribution

It introduces the concept of positive representations in Riesz spaces, classifies finite dimensional Archimedean cases, and explores their irreducibility and induction properties, highlighting differences from classical theory.

Findings

Finite dimensional positive Archimedean representations are order direct sums of indecomposables.

Character theory does not apply to positive representations.

Induction and imprimitivity systems behave differently in the ordered setting.

Abstract

In this paper, which is part of a study of positive representations of locally compact groups in Banach lattices, we initiate the theory of positive representations of finite groups in Riesz spaces. If such a representation has only the zero subspace and possibly the space itself as invariant principal bands, then the space is Archimedean and finite dimensional. Various notions of irreducibility of a positive representation are introduced and, for a finite group acting positively in a space with sufficiently many projections, these are shown to be equal. We describe the finite dimensional positive Archimedean representations of a finite group and establish that, up to order equivalence, these are order direct sums, with unique multiplicities, of the order indecomposable positive representations naturally associated with transitive $G$-spaces. Character theory is shown to break down for positive representations. Induction and systems of imprimitivity are introduced in an ordered context, where the multiplicity formulation of Frobenius reciprocity turns out not to hold.