Paper detail

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

We build a bijection between the set $\sttiltΛ$ of isomorphism classes of basic support $τ$-tilting modules over the Auslander algebra $Λ$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on $\sttiltΛ$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\tiltΛ$ of isomorphism classes of basic tilting $Λ$-modules and the symmetric group $\mathfrak{S}_n$ 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 $-\otimes_ΛΓ$ induces a bijection between $\sttiltΛ\to\sttiltΓ$. This recover Mizuno's bijection $\mathfrak{S}_{n+1}\to\sttiltΓ$ for type $A_n$.