An expansion formula for type A and Kronecker quantum cluster algebras

We introduce an expansion formula for elements in quantum cluster algebras associated to type A and Kronecker quivers with principal quantization. Our formula is parametrized by perfect matchings of snake graphs as in the classical case. In the Kronecker type, the coefficients are q-powers whose exp...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series A 2020-04, Vol.171, p.105132, Article 105132
Main Authors: Çanakçı, İlke, Lampe, Philipp
Format: Article
Language:English
Subjects:
Arcs
Lattices
Mathematical physics
Non-commutative rings
Perfect matchings
Quantum cluster algebras
Snake graphs
Stembridge phenomenon
Surface cluster algebras
Triangulations
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Summary:We introduce an expansion formula for elements in quantum cluster algebras associated to type A and Kronecker quivers with principal quantization. Our formula is parametrized by perfect matchings of snake graphs as in the classical case. In the Kronecker type, the coefficients are q-powers whose exponents are given by a weight function induced by the lattice of perfect matchings. As an application, we prove that a reflectional symmetry on the set of perfect matchings satisfies Stembridge's q=−1 phenomenon with respect to the weight function. Furthermore, we discuss a relation of our expansion formula to generating functions of BPS states.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2019.105132