Triangulations of prisms and preprojective algebras of type $A$

TL;DR

This paper establishes a correspondence between triangulations of prisms and certain algebraic structures called two-term silting complexes over preprojective algebras of type A, linking geometric and algebraic combinatorics.

Contribution

It introduces a novel bijection between prism triangulations and two-term silting complexes, connecting geometric triangulations with algebraic mutations in preprojective algebras.

Findings

Bijection between indecomposable two-term presilting complexes and internal n-simplices in prisms.

Triangulations of prisms correspond to two-term silting complexes.

Bistellar flips correspond to mutations of silting complexes.

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.