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.
Keywords
Cite
@article{arxiv.1409.4723,
title = {Initial-seed recursions and dualities for d-vectors},
author = {Nathan Reading and Salvatore Stella},
journal= {arXiv preprint arXiv:1409.4723},
year = {2026}
}
Comments
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