Cluster g\mathbf g-vector fan conjecture

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For each vertex vv of the exchange graph, let Cone⁡(v)\operatorname{Cone}(v) be the cone in V∗V^* spanned by the weights g(xev)\mathbf g(x^v_e) for e∈I(v)e\in I(v). A fan is a collection of cones closed under faces in which any two cones meet in a face of each. Cluster g\mathbf g-vector fan conjecture. The collection {Cone⁡(v):v∈Ex0(B)}\{\operatorname{Cone}(v):v\in\mathrm{Ex}_0(B)\} is a fan and the map v↦Cone⁡(v)v\mapsto\operatorname{Cone}(v) is injective. The source identifies this with a restatement of the cited conjecture and gives no general resolution.

References

Primary source

Nathan Reading and David E Speyer, “Combinatorial frameworks for cluster algebras”, arXiv:1111.2652 (2026).

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