Double-dimer configurations and quivers of dP3 (del Pezzo) type

Abstract: In this talk, I will describe our work extending combinatorial interpretations for so called toric cluster variables as was previously studied by myself and Tri Lai. In [LM 2017] and [LM 2020], most toric cluster variables were shown to have Laurent expansions agreeing with partition functions of dimers on subgraphs cut out by six-sided contours. However, the case of cluster variables parameterized by six-sided contours with a self-intersection eluded our techniques. In this talk we discuss our research rectifying this issue by using Helen Jenne’s condensation results for the double-dimer model [J 2019]. While we focus on quivers of dP3 type of Model 1 and Model 4, we anticipate our techniques will extend to certain additional cluster algebras related to brane tilings.  This is joint work with Helen Jenne and Tri Lai.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)