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The denominator-vector injectivity conjecture for cluster monomials

Let Σt0\Sigma_{t_0} be an initial seed, and let da;tB0;t0\mathbf{d}_{\mathbf{a};t}^{B^0;t_0} denote the denominator vector of a cluster monomial at vertex tt. Denominator-vector injectivity conjecture. Distinct cluster monomials have distinct denominator vectors with respect to Σt0\Sigma_{t_0}. In particular, for each fixed tt, the map

a⟼da;tB0;t0\mathbf{a}\longmapsto\mathbf{d}_{\mathbf{a};t}^{B^0;t_0}

is an embedding of Z≥0n\mathbb{Z}_{\geq0}^n into Zn\mathbb{Z}^n, so the denominator vectors of the cluster variables at tt form a basis of Qn\mathbb{Q}^n. The paper records finite-type evidence via the cited proposition, but the general statement remains open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Denominator-vector injectivity conjecture for cluster monomials

    A cluster monomial is a monomial in cluster variables from one cluster, and its denominator vector is obtained from the denominator vectors of its variables with multiplicities. Denominator-vector injectivity conjecture for cluster monomials. Different cluster monomials have different denominator vectors. The source later states that this holds for finite Cartan type, but gives no general resolution.

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

Sources & referencesView supporting material

Primary source

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

cluster-algebrasdenominator-vectors

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