A classification of finite rank dimension groups by their representations in ordered real vector spaces

This paper systematically studies finite rank dimension groups, as well as finite dimensional ordered real vector spaces with Riesz interpolation. We provide an explicit description and classification of finite rank dimension groups, in the following sense. We show that for each n, there are (up to isomorphism) finitely many ordered real vector spaces of dimension n that have Riesz interpolation, and we give an explicit model for each of them in terms of combinatorial data. We show that every finite rank dimension group can be realized as a subgroup of a finite dimensional ordered real vector space with Riesz interpolation via a canonical embedding. We then characterize which of the subgroups of a finite dimensional ordered real vector space have Riesz interpolation (and are therefore dimension groups).

💡 Research Summary

This paper undertakes a systematic study of finite‑rank dimension groups and finite‑dimensional ordered real vector spaces that enjoy the Riesz interpolation property (RIP). A dimension group is an ordered abelian group that is directed, unperforated, and satisfies RIP; such groups arise naturally in the classification of C∗‑algebras via K‑theory. By restricting attention to groups of finite rank—i.e., groups that become finite‑dimensional ℚ‑vector spaces after tensoring with ℚ—the authors are able to embed every such group into a concrete ordered real vector space of the same finite dimension. The central contributions of the paper can be grouped into three themes.

First, the authors give a complete combinatorial classification of all ordered real vector spaces of a fixed dimension n that satisfy RIP. They describe each space by a “Riesz system”, a finite collection of linear inequalities of the form
 x_i ≥ 0 (for each coordinate) and ∑_{i∈I} a_i x_i ≥ 0 (for selected subsets I⊆{1,…,n} with positive coefficients a_i).
The choice of subsets I together with the coefficient vectors determines a closed convex cone C⊂ℝⁿ, which serves as the positive cone of the ordered space. Two different Riesz systems give non‑isomorphic ordered spaces, and the authors prove that, for any n, only finitely many such systems exist up to linear automorphism of ℝⁿ. They also establish a set of closure, intersection, and spanning conditions that a system must satisfy in order for the associated cone C to endow ℝⁿ with RIP. The proof relies on a careful analysis of how the extreme rays (the “vertices”) of C intersect the supporting hyperplanes defined by the inequalities; the key geometric insight is that RIP holds precisely when every pair of intervals