Triangulations of Prisms and Preprojective Algebras of Type A

Author:

Iyama Osamu1,Williams Nicholas J2

Affiliation:

1. Graduate School of Mathematical Sciences , The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

2. Department of Mathematics and Statistics , Fylde College, Lancaster University, Lancaster, LA1 4YF, UK

Abstract

Abstract 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.

Publisher

Oxford University Press (OUP)

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