### 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 language | English |
---|---|

Article number | 105132 |

Pages (from-to) | 1-30 |

Number of pages | 30 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 171 |

Early online date | 31 Oct 2019 |

DOIs | |

Publication status | E-pub ahead of print - 31 Oct 2019 |

### Fingerprint

### Keywords

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

### Cite this

*Journal of Combinatorial Theory. Series A*,

*171*, 1-30. [105132]. https://doi.org/10.1016/j.jcta.2019.105132

}

*Journal of Combinatorial Theory. Series A*, vol. 171, 105132, pp. 1-30. https://doi.org/10.1016/j.jcta.2019.105132

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

Research output: Contribution to Journal › Article › Academic › peer-review

TY - JOUR

T1 - An expansion formula for type A and Kronecker quantum cluster algebras

AU - Çanakçı, İlke

AU - Lampe, Philipp

PY - 2019/10/31

Y1 - 2019/10/31

N2 - 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.

AB - 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.

KW - Arcs

KW - Lattices

KW - Mathematical physics

KW - Non-commutative rings

KW - Perfect matchings

KW - Quantum cluster algebras

KW - Snake graphs

KW - Stembridge phenomenon

KW - Surface cluster algebras

KW - Triangulations

UR - http://www.scopus.com/inward/record.url?scp=85074159881&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85074159881&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2019.105132

DO - 10.1016/j.jcta.2019.105132

M3 - Article

VL - 171

SP - 1

EP - 30

JO - Journal of Combinatorial Theory

JF - Journal of Combinatorial Theory

SN - 0097-3165

M1 - 105132

ER -