Initial-seed recursions and dualities for d-vectors

Abstract

We present an initial-seed-mutation formula for d-vectors of cluster variables in a cluster algebra. We also give two rephrasings of this recursion: one as a duality formula for d-vectors in the style of the g-vectors/c-vectors dualities of Nakanishi and Zelevinsky, and one as a formula expressing the highest powers in the Laurent expansion of a cluster variable in terms of the d-vectors of any cluster containing it. We prove that the initial-seed-mutation recursion holds in a varied collection of cluster algebras, but not in general. We conjecture further that the formula holds for source-sink moves on the initial seed in an arbitrary cluster algebra, and we prove this conjecture in the case of surfaces.

21 Pages, 20 Figures. Version 2: Expanded introduction, other minor expository changes. Version 3: Very minor corrections. Final version to appear in the Pacific Journal of Mathematics

References in corpus (5)

• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• Denominator vectors and compatibility degrees in cluster algebras of finite type
• Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry
• Universal geometric coefficients for the once-punctured torus
• Some Consequences of Categorification

Cited by in corpus (7)

• Duality between Final-Seed and Initial-Seed Mutations in Cluster Algebras
• The valuation pairing on an upper cluster algebra
• Some Consequences of Categorification
• Positivity of denominator vectors of cluster algebras
• Study on cluster algebras via abstract pattern and two conjectures on d-vectors and g-vector
• Compatibility degree of cluster complexes
• Difference equations arising from cluster algebras