Abstract

In this paper, we prove some combinatorial results on generalized cluster algebras . To be more precise, we prove that (i) the seeds of a generalized cluster algebra A ( S ) whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of A ( S ) ; (ii) there exists a bijection from the set of cluster variables of a generalized cluster algebra to the set of cluster variables of another generalized cluster algebra, if their initial exchange matrices satisfy a mild condition. Moreover, this bijection preserves the set of clusters of these two generalized cluster algebras. As applications of the second result, we prove some properties of the components of the d -vectors of a generalized cluster algebra and we give a characterization for the clusters of a generalized cluster algebra.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.