Characterization of group-derived weighted flow configurations

Characterize which weighted flow configurations arise from a group, thereby determining the image of the representation-theoretic construction that produces the unit-weight, mixed-order, induced, and recombined configurations.

Background

The paper identifies several orbifold-derived configurations with established objects such as cut configurations, group-based phylogenetic models, and binary hierarchical models. Weighted configurations arising from higher-dimensional irreducible representations and from real or quaternionic recombination fall outside those frameworks.

The authors describe a general weighted-flow perspective but do not provide an abstract characterization of which such configurations can actually arise from a group. Establishing this characterization would turn the listed weighted examples and their gluing trees into instances of a well-defined class.

References

The content is in the other direction, and it is the question this note ends on: Which weighted flow configurations arise from a group?

— An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them  (2609.02630 - Marín, 2 Sep 2026) in Section 7, “Where the weights take it outside”