Exceptional sequences and spherical modules for the Auslander algebra of $k[x]/(x^t)$

TL;DR

This paper classifies spherical modules and exceptional sequences over the Auslander algebra of k[x]/(x^t), and describes their symmetries via braid group actions related to spherical twists and mutations.

Contribution

It provides a complete classification of exceptional sequences and spherical modules for this algebra, and links their symmetries to braid group actions.

Findings

Every exceptional sequence is obtained by spherical twists from a standard sequence.

The categorification of group actions leads to two braid group actions: spherical twists and mutations.

The classification enhances understanding of the module category structure for the algebra.

Abstract

We classify spherical modules and full exceptional sequences of modules over the Auslander algebra of $k[x]/(x^t)$. We categorify the left and right symmetric group actions on these exceptional sequences to two braid group actions: of spherical twists along simple modules, and of right mutations. In particular, every such exceptional sequence is obtained by spherical twists from a standard sequence, and likewise for right mutations.