An expansion formula for type A and Kronecker quantum cluster algebras

İlke Çanakçı, Philipp Lampe

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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.

Original languageEnglish
Article number105132
Pages (from-to)1-30
Number of pages30
JournalJournal of Combinatorial Theory. Series A
Early online date31 Oct 2019
Publication statusPublished - Apr 2020


  • Arcs
  • Lattices
  • Mathematical physics
  • Non-commutative rings
  • Perfect matchings
  • Quantum cluster algebras
  • Snake graphs
  • Stembridge phenomenon
  • Surface cluster algebras
  • Triangulations


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