A simple algebraic proof of the non-transitivity of the braid group action on full exceptional sequences

TL;DR

This paper offers a straightforward algebraic proof demonstrating that the braid group action on full exceptional sequences in a triangulated category is not transitive, confirming the existence of infinitely many orbits.

Contribution

It introduces a simple algebraic proof of the non-transitivity of the braid group action, simplifying previous geometric approaches.

Findings

Braid group action on full exceptional sequences is not transitive.

There are infinitely many orbits under this action.

The proof is algebraic, avoiding complex geometric models.

Abstract

Recently, Chang--Haiden--Schroll shows that the braid group action on full exceptional collections in a triangulated category is not transitive but has infinitely many orbits in general. Their proof is based on a geometric model and the theory of branched coverings such as Birman--Hilden theory. This paper provides a simple algebraic proof of their theorem.