Relation between \(f\)-vectors and \(d\)-vectors in cluster algebras of finite type or rank 2

We study \(f\)-vectors, which are the maximal degree vectors of \(F\)-polynomials in cluster algebra theory. For a cluster algebra is of finite type, we find that positive \(f\)-vectors correspond with \(d\)-vectors, which are exponent vectors of denominators of cluster variables. Furthermore, using...

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Bibliographic Details
Published in:arXiv.org 2021-08
Main Author: Gyoda, Yasuaki
Format: Article
Language:English
Subjects:
Algebra
Clusters
Mathematical analysis
Polynomials
Vectors (mathematics)
Online Access:Get full text
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Summary:We study \(f\)-vectors, which are the maximal degree vectors of \(F\)-polynomials in cluster algebra theory. For a cluster algebra is of finite type, we find that positive \(f\)-vectors correspond with \(d\)-vectors, which are exponent vectors of denominators of cluster variables. Furthermore, using this correspondence and properties of \(d\)-vectors, we prove that cluster variables in a cluster are uniquely determined by their \(f\)-vectors when the cluster algebra is of finite type or rank \(2\).
ISSN:2331-8422
DOI:10.48550/arxiv.1904.00779