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

TL;DR

This paper establishes a bijection between support $ au$-tilting modules over the Auslander algebra of $K[x]/(x^n)$ and the symmetric group $rak{S}_{n+1}$, connecting algebraic structures with combinatorial groups.

Contribution

It constructs a new bijection linking support $ au$-tilting modules to symmetric groups and relates these modules over different algebras via tensor functors, extending known results.

Findings

Bijection between support $ au$-tilting modules and symmetric group $rak{S}_{n+1}$

Restriction to tilting modules and symmetric group $rak{S}_n$

Tensor functor induces bijection between modules over $ ext{Auslander algebra}$ and $ ext{preprojective algebra}$

Abstract

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 $A_n$.