Crystal Structure of Upper Cluster Algebras

We describe the upper seminormal crystal structure for the \(\mu\)-supported \(\delta\)-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster \(\mc{X}\)-variety. We show that the crystal structure can be algebraical...

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Bibliographic Details
Published inarXiv.org
Main Author Fei, Jiarui
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 15.09.2023
Subjects
Algebra
Apexes
Clusters
Crystal structure
Mathematics - Combinatorics
Mathematics - Representation Theory
Mathematics - Rings and Algebras
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Summary:We describe the upper seminormal crystal structure for the \(\mu\)-supported \(\delta\)-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster \(\mc{X}\)-variety. We show that the crystal structure can be algebraically lifted to the generic basis of the upper cluster algebra. This can be viewed as an additive categorification of the crystal structure arising from cluster algebras. We introduce the biperfect bases and the strong biperfect bases in the cluster algebra setting and give a description of all (strong) biperfect bases, which are parametrized by lattice points in a product of explicit polyhedral sets. We illustrate this theory from classical examples and new examples.
ISSN:2331-8422
DOI:10.48550/arxiv.2309.08326