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July, 2020 Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$
Osamu IYAMA, Xiaojin ZHANG
J. Math. Soc. Japan 72(3): 731-764 (July, 2020). DOI: 10.2969/jmsj/75117511


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).


Download 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.


Received: 4 May 2016; Revised: 27 October 2018; Published: July, 2020
First available in Project Euclid: 15 November 2019

zbMATH: 07257208
MathSciNet: MR4125843
Digital Object Identifier: 10.2969/jmsj/75117511

Primary: 16G10
Secondary: 16E10

Keywords: $\tau$-tilting module , Auslander algebra , preprojective algebra , Symmetric group , tilting module

Rights: Copyright © 2020 Mathematical Society of Japan


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Vol.72 • No. 3 • July, 2020
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