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Relatively dominated representations give a common generalization ofgeometrically finiteness in rank one on the one hand, and the Anosov conditionwhich serves as a higher-rank analogue of convex cocompactness on the other.This note proves three results about these representations.  Firstly, we remove the technical assumption of quadratic gaps involved in theoriginal definition. Secondly, we give a characterization using eigenvaluegaps, providing a relative analogue of a result of Kassel-Potrie for Anosovrepresentations. Thirdly, we formulate characterizations in terms of singularvalue or eigenvalue gaps combined with limit maps, in the spirit ofGu\'eritaud-Guichard-Kassel-Wienhard for Anosov representations, and use themto show that inclusion representations of certain groups playing weak ping-pongand positive representations in the sense of Fock-Goncharov are relativelydominated.

Relatively dominated representations from eigenvalue gaps and limit maps