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

Abstract

We build a bijection between the set $\sttiltΛ$ 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 $\sttiltΛ$ and the left order on ${\mathfrak{S}}_{n+1}$. This restricts to the bijection between the set $\tiltΛ$ of isomorphism classes of basic tilting $\Lambda$-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 $\sttiltΛ\to\sttiltΓ$. This recover Mizuno's bijection $\mathfrak{S}_{n+1}\to\sttiltΓ$ for type $A_n$.

References in corpus (4)

• $\tau$-tilting finite algebras, bricks and $g$-vectors
• Classification of two-term tilting complexes over Brauer graph algebras
• Lattice structure of Weyl groups via representation theory of preprojective algebras
• Tilting modules over Auslander-Gorenstein Algebras

Cited by in corpus (6)

• Lattice theory of torsion classes: Beyond $\tau$-tilting theory
• Silting objects
• Weak orders on symmetric groups and posets of support $\tau$-tilting modules
• A new characterization of Auslander algebras
• On the tilting complexes for the Auslander algebra of the truncated polynomial ring
• On mutation of τ-tilting modules