A classification of $n$-representation infinite algebras of type \~A

We classify $n$-representation infinite algebras $\Lambda$ of type \~A. This type is defined by requiring that $\Lambda$ has higher preprojective algebra $\Pi_{n+1}(\Lambda) \simeq k[x_1, \ldots, x_{n+1}] \ast G$, where $G \leq \operatorname{SL}_{n+1}(k)$ is finite abelian. For the classification, we group these algebras according to a more refined type, and give a combinatorial characterisation of these types. This is based on so-called height functions, which generalise the height function of a perfect matching in a Dimer model. In terms of toric geometry and McKay correspondence, the types form a lattice simplex of junior elements of $G$. We show that all algebras of the same type are related by iterated $n$-APR tilting, and hence are derived equivalent. By disallowing certain tilts, we turn this set into a finite distributive lattice, and we construct its maximal and minimal elements.