Classifying 𝜏-tilting modules over the Auslander algebra of 𝐾[𝑥]/(𝑥^𝑛)

Abstract

We build a bijection between the set s𝜏-tilt Λ of isomorphism classes of basic support 𝜏-tilting modules over the Auslander algebra Λ of 𝐾[𝑥]/(𝑥^𝑛) and the symmetric group 𝔖_𝑛+1, which is an anti-isomorphism of partially ordered sets with respect to the generation order on s𝜏-tilt Λ and the left order on 𝔖_𝑛+1. This restricts to the bijection between the set tilt Λ of isomorphism classes of basic tilting Λ-modules and the symmetric group 𝔖_𝑛 due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra Γ of Dynkin type 𝐴_𝑛 as a factor algebra of Λ, we show that the tensor functor −⊗_Λ Γ induces a bijection between s𝜏-tilt Λ → s𝜏-tilt Γ. This recover Mizuno's anti-isomorphism 𝔖_𝑛+1 → s𝜏-tilt Γ of posets for type 𝐴_𝑛.