The 3-Preprojective Algebras Of Type $\tilde{A}$

Let $G\le {\mathrm{SL}}_{n+1}(\mathbb{C})$ act on $R=\mathbb{C}[X_1,\dots ,X_{n+1}]$ by change of variables. Then, the skew-group algebra $R\ast G$ is bimodule $(n+1)$-Calabi-Yau. Under certain circumstances, the algebra admits a locally finite-dimensional grading of Gorenstein parameter $1$, in which case it is the $(n+1)$-preprojective algebra of its $n$-representation infinite degree $0$ piece, as defined by Herschend, Iyama and Oppermann. If the group $G$ is abelian, the $(n+1)$-preprojective algebra is said to be of type $\tilde{A}$. For a given group $G$, it is not obvious whether $R\ast G$ admits such a grading making it into an $(n+1)$-preprojective algebra. We study the case when $n=2$ and $G$ is abelian. We give an explicit classification of groups such that $R\ast G$ is $3$-preprojective by constructing such gradings. This is possible as long as $G$ is not a subgroup of ${\mathrm{SL}}_2(\mathbb{C})$ and not $C_2\times C_2$. For a fixed $G$, the algebra $R\ast G$ admits different $3$-preprojective gradings, so we associate a type to a grading and classify all types. Then we show that gradings of the same type are related by a certain kind of mutation. This gives a classification of $2$-representation infinite algebras of type $\tilde{A}$. The involved quivers are those arising from hexagonal dimer models on the torus, and the gradings we consider correspond to perfect matchings on the dimer, or equivalently to periodic lozenge tilings of the plane. Consequently, we classify these tilings up to flips, which correspond to the mutation we consider.