A classification of n-representation infinite algebras of type Ã

We classify n -representation infinite algebras Λ Λ of type à A ~ . This type is defined by requiring that Λ Λ has higher preprojective algebra Π ₙ₊₁(Λ ) k[x₁, … , xₙ₊₁] *G Π n + 1 ( Λ ) ≃ k [ x 1 , … , x n + 1 ] ∗ G , where G ≤ \,SL\,ₙ₊₁(k) G ≤ 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.