TY - UNPB

T1 - Triangulations of prisms and preprojective algebras of type $A$

AU - Iyama, Osamu

AU - Williams, Nicholas J.

N1 - 20 pages, 5 figures. v2: fixed typos

PY - 2022/8/27

Y1 - 2022/8/27

N2 - 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.

AB - 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.

KW - math.RT

KW - math.CO

KW - 05E10, 16G20, 52B12

M3 - Preprint

BT - Triangulations of prisms and preprojective algebras of type $A$

ER -