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

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.