An expansion formula for type A and Kronecker quantum cluster algebras

İlke Çanakçı, Philipp Lampe

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

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
Volume171
Early online date31 Oct 2019
DOIs
Publication statusE-pub ahead of print - 31 Oct 2019

Fingerprint

Cluster Algebra
Quantum Algebra
Perfect Matching
Algebra
Weight Function
Reflectional symmetry
Snakes
Quiver
Generating Function
Quantization
Exponent
Coefficient
Graph in graph theory

Keywords

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

Cite this

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An expansion formula for type A and Kronecker quantum cluster algebras. / Çanakçı, İlke; Lampe, Philipp.

In: Journal of Combinatorial Theory. Series A, Vol. 171, 105132, 01.04.2020, p. 1-30.

Research output: Contribution to JournalArticleAcademicpeer-review

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