# On the tilting complexes for the Auslander algebra of the truncated polynomial ring

## TL;DR

This paper establishes a bijection between tilting complexes of the Auslander algebra of a truncated polynomial ring and a product of integers and braid group elements, revealing their mutation structure and automorphism group.

## Contribution

It introduces a new classification of tilting complexes for this algebra using braid group actions and describes its derived Picard group structure.

## Key findings

- Bijection between tilting complexes and Z x B
- Mutation components parametrized by Z with braid group action
- Derived Picard group characterized as a product involving the braid group

## Abstract

We give a bijection between the tilting complexes in the bounded homotopy category of the Auslander algebra of the truncated polynomial ring and ZxB where B is the Artin braid goup of type A with n-1 generators. The tilting complexes have mutation components parametrized by Z and each component has a natural faithful and transitive operation of B. This also implies that the derived Picard group of this algebra is isomorphic to the direct product of its outer isomorphism group and ZxB. This work is to be seen as a continuation of the work of Geuenich and an application of the work of Aihara and Mizuno on tilting complexes of preprojective algebras of Dynkin type.