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

Abstract

We build a bijection between the set ${\rm s}\tau\mbox{-tilt}\hspace{.01in} \Lambda$ of isomorphism classes of basic support $\tau$-tilting modules over the Auslander algebra $\Lambda$ 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 ${\rm s}\tau\mbox{-tilt}\hspace{.01in} \Lambda$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set ${\rm tilt}\hspace{.01in} \Lambda$ of isomorphism classes of basic tilting $\Lambda$-modules and the symmetric group $\mathfrak{S}_n$ due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra $\Gamma$ of Dynkin type $A_n$ as a factor algebra of $\Lambda$, we show that the tensor functor $-\otimes_{\Lambda} \Gamma$ induces a bijection between ${\rm s}\tau\mbox{-tilt}\hspace{.01in} \Lambda\to {\rm s}\tau\mbox{-tilt}\hspace{.01in} \Gamma$. This recover Mizuno's anti-isomorphism $\mathfrak{S}_{n+1} \to {\rm s}\tau\mbox{-tilt}\hspace{.01in} \Gamma$ of posets for type $A_n$.