Triangulations of Prisms and Preprojective Algebras of Type A

Abstract

We show that isomorphism classes of indecomposable $\tau $-rigid pairs over $\Pi _{n}$, the preprojective algebra of $A_{n}$, are in bijection with internal $n$-simplices in the prism $\Delta _{n} \times \Delta _{1}$, the product of an $n$-simplex with a 1-simplex. We show further that this induces a bijection between triangulations of $\Delta _{n} \times \Delta _{1}$ and basic support $\tau $-tilting pairs over $\Pi _{n}$ such that bistellar flips of triangulations correspond to mutations of support $\tau $-tilting pairs. These bijections are shown to be compatible with the known bijections involving the symmetric group.