Reading's separation conjecture for cluster variables

Let A\mathcal A be a cluster algebra. For cluster variables u=xi;tu=x_{i;t} and u=xj;tu'=x_{j;t'}, write gk;usg_{k;u}^s and gk;usg_{k;u'}^s for the kkth components of their gg-vectors at a vertex ss of the exchange tree. The variables are sign-coherent when

gk;usgk;us0g_{k;u}^s g_{k;u'}^s\geq 0

for every vertex ss and every k[1,n]k\in[1,n].

Reading's separation conjecture. The variables uu and uu' are contained in the same cluster if and only if they are sign-coherent.

This reformulates Reading's conjectured separation property for cluster variables in universal geometric cluster algebras. The supplied text does not state whether the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Peigen Cao, “F-invariant and E-invariant”, arXiv:2503.06605 (2025).

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