Existence of variation maps between torus algebras of similar seeds

Determine whether, for arbitrary pairs of similar seeds (beyond principal coefficient cases), there exists a variation map var: LP→LP′—a k-algebra homomorphism sending each cluster variable x_k to a frozen-factor multiple p_k·x′_k and satisfying var(x^{col_k B})=(x′)^{col_k B′} for all unfrozen indices k. Establish necessary and sufficient conditions for the existence of such maps or explicit constructions.

Background

Variation maps are central to the base change technique developed in the paper, allowing the propagation of structures and properties between cluster algebras associated with similar seeds. While principal-coefficient seeds always admit such maps, the general existence for arbitrary pairs of similar seeds is not known.

Resolving existence criteria for variation maps would broaden the applicability of base change isomorphisms and enable systematic transfer of bases and cluster structures across related seeds.