MathDBThe linear-independence conjecture for cluster monomials
The linear-independence conjecture for cluster monomials
Let be a cluster algebra over ground ring . A cluster monomial is a monomial in cluster variables belonging to one cluster. Cluster-monomial linear-independence conjecture. Cluster monomials in any cluster algebra are linearly independent over the ground ring. This is a basic structural conjecture for cluster algebras. It is proved for cluster algebras of finite type, while the general case 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.
Linear independence conjecture for cluster monomials
Let be a cluster algebra. A cluster monomial is a monomial in cluster variables, all of which belong to the same cluster. Cluster-monomial linear independence conjecture. The cluster monomials are linearly independent. The conjecture is known for cluster algebras defined by quivers, but the source does not establish it in general.
source: Lauren K. Williams, “Cluster algebras: an introduction”, arXiv:1212.6263 (2013).
Sources & referencesView supporting material
Primary source
Sergey Fomin and Andrei Zelevinsky, “Cluster algebras IV: Coefficients”, arXiv:math/0602259 (2006).
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