Triangulations of Prisms and Preprojective Algebras of Type A

We show that isomorphism classes of indecomposable τ-rigid pairs over Π ₙ, the preprojective algebra of Aₙ, are in bijection with internal n-simplices in the prism Δ ₙ × Δ ₁, the product of an n-simplex with a 1-simplex. We show further that this induces a bijection between triangulations of Δ ₙ × Δ ₁ and basic support τ-tilting pairs over Π ₙ such that bistellar flips of triangulations correspond to mutations of support τ-tilting pairs. These bijections are shown to be compatible with the known bijections involving the symmetric group.