TY - JOUR

T1 - Triangulations of Prisms and Preprojective Algebras of Type A

AU - Iyama, Osamu

AU - Williams, Nicholas J

PY - 2024/4/16

Y1 - 2024/4/16

N2 - 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.

AB - 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.

U2 - 10.1093/imrn/rnae059

DO - 10.1093/imrn/rnae059

M3 - Journal article

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

M1 - rnae059

ER -