Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$
Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$
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DOI：
10.2969/jmsj/75117511
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发表时间：
2016-02
影响因子：
0.7
通讯作者：
O. Iyama;Xiaojin Zhang
中科院分区：
文献类型：
--
作者：
O. Iyama;Xiaojin Zhang
我们在$K[x]/（x^n）$的Auslander代数$\Lambda$上的基本支撑$\tau$-倾斜模的同构类的集合$\sttilt\Lambda$和对称群$\mathfrak{S}_{n+1}$之间建立了一个双射，它是偏序集合关于$\sttilt\Lambda$上的生成阶和$\mathfrak{S}_{n+1}$上的左阶的反同构.这限制了基本倾斜$\Lambda$-模的同构类的集合$\tilt\Lambda$与对称群$\mathfrak{S}_n$之间的双射，由于Br\"{u}stle，Hille，Ringel和R\"{o}hrle。将Dynkin型预投射代数$\Gamma$设为$\Lambda$的因子代数，证明了张量函子$-\otimes_{\Lambda}\Gamma$诱导$\sttilt\Lambda\到\sttilt\Gamma$之间的双射.这恢复了Mizuno的双射$\mathfrak{S}_{n+1}\到\sttilt\Gamma$，类型为$A_n$。
We build a bijection between the set $\sttilt\Lambda$ 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 $\sttilt\Lambda$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\tilt\Lambda$ of isomorphism classes of basic tilting $\Lambda$-modules and the symmetric group $\mathfrak{S}_n$ due to Br\"{u}stle, Hille, Ringel and R\"{o}hrle. 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 $\sttilt\Lambda\to\sttilt\Gamma$. This recover Mizuno's bijection $\mathfrak{S}_{n+1}\to\sttilt\Gamma$ for type $A_n$.