We classify n-representation infinite algebras \(\Lambda \) of type \(\tilde{A}\) . This type is defined by requiring that \(\Lambda \) has higher preprojective algebra \(\Pi _{n+1}(\Lambda ) \simeq k[x_1, \ldots , x_{n+1}] *G\) , where \(G \le {{\,\textrm{SL}\,}}_{n+1}(k)\) 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 the 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.
