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

We classify spherical modules and full exceptional sequences of modules over the Auslander algebra of $k[x]\mathrm{/}(x^t)$. 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.