Tilting modules for the Auslander algebra of $K[x]/(x^n)$

TL;DR

This paper establishes a correspondence between tilting modules for a specific algebra and certain elements in the braid group, revealing the finiteness of such modules.

Contribution

It constructs an isomorphism linking tilting modules of the Auslander algebra of $K[x]/(x^n)$ to rational permutation braids, showing their finite nature.

Findings

Finitely many tilting modules for the algebra.

Isomorphism with rational permutation braids.

Structural insight into module classification.

Abstract

We construct an isomorphism between the partially ordered set of tilting modules for the Auslander algebra of $K[x]/(x^n)$ and the interval of rational permutation braids in the braid group on $n$ strands. Hence, there are only finitely many tilting modules.