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 $A_n$.
Funding Statement
The first author was supported by JSPS Grant-in-Aid for Scientific Research (B) 24340004, (C) 23540045 and (S) 15H05738. The second author was supported by NSFC (Nos.11571164, 11671174) and Jiangsu Government Scholarship for Overseas Studies (JS-2014-352).
Citation
Osamu IYAMA. Xiaojin ZHANG. "Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$." J. Math. Soc. Japan 72 (3) 731 - 764, July, 2020. https://doi.org/10.2969/jmsj/75117511
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