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 |
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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 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 -