Triangulations of prisms and preprojective algebras of type $A$

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.

20 pages, 5 figures. v2: fixed typos