The sign-coherence conjecture for denominator vectors

Let di;tB0;t0\mathbf{d}_{i;t}^{B^0;t_0} be the denominator vectors of the cluster variables at a vertex tt relative to the initial vertex t0t_0. A collection of vectors is sign-coherent if, in each coordinate, all vectors have either nonnegative or nonpositive entries. Sign-coherence conjecture for denominator vectors. For every tt and t0t_0, the vectors d1;tB0;t0,,dn;tB0;t0\mathbf{d}_{1;t}^{B^0;t_0},\dots,\mathbf{d}_{n;t}^{B^0;t_0} are sign-coherent. The paper notes that this follows from the preceding denominator-vector conjecture and proves it on the bipartite belt; it remains open in general.

Sources & referencesView supporting material

Primary source

Sergey Fomin and Andrei Zelevinsky, “Cluster algebras IV: Coefficients”, arXiv:math/0602259 (2006).

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