Triangulations of prisms and preprojective algebras of type $A$

Iyama, Osamu and Williams, Nicholas J. (2022) Triangulations of prisms and preprojective algebras of type $A$. Other. UNSPECIFIED.

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Abstract

We show that indecomposable two-term presilting complexes 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 two-term silting complexes over $\Pi_{n}$ such that bistellar flips of triangulations correspond to mutations of two-term silting complexes. These bijections are shown to compatible with the known bijections involving the symmetric group.