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

Atsuki Nakago, Atsushi Takahashi

TL;DR

The paper addresses the non-transitivity of the braid group action on full exceptional collections in derived categories. It provides an algebraic proof of CHS's result, using spherical objects and Seidel–Thomas twists to generate infinitely many orbits, first in the μ=4 case and then for μ≥5 by reduction. The key contributions are constructing explicit 3-spherical objects and showing they produce distinct braid-orbit representatives, and by reduction, extending the non-transitivity to general μ. This strengthens understanding of mutation dynamics in triangulated categories and the structure of full exceptional collections beyond geometric methods.

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.

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

TL;DR

The paper addresses the non-transitivity of the braid group action on full exceptional collections in derived categories. It provides an algebraic proof of CHS's result, using spherical objects and Seidel–Thomas twists to generate infinitely many orbits, first in the μ=4 case and then for μ≥5 by reduction. The key contributions are constructing explicit 3-spherical objects and showing they produce distinct braid-orbit representatives, and by reduction, extending the non-transitivity to general μ. This strengthens understanding of mutation dynamics in triangulated categories and the structure of full exceptional collections beyond geometric methods.

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.

Paper Structure

This paper contains 7 sections, 9 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Full exceptional collections
  4. Group actions on the set of full exceptional collections
  5. Algebraic proof of the non-transitivity
  6. The case $\mu=4$
  7. The case $\mu\geq 5$

Key Result

Theorem 1.1

Let $\mu\ge 4$ and $(Q, I)$ the quiver with relations given by Let ${\mathbb C} Q/I$ be the path algebra of $(Q,I)$ and ${\mathcal{D}}^{b}{\rm mod} ({\mathbb C} Q/I)$ the bounded derived category of finite dimensional ${\mathbb C} Q/I$-modules. The $B_\mu\ltimes {\mathbb Z}^\mu$-action on the set of full exceptional collections in ${\mathcal{D}}^b{\rm mod}({\ma

Theorems & Definitions (18)

  • Theorem 1.1: CHS
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: BK
  • Remark 2.4
  • Definition 2.5: ST
  • Proposition 2.6: ST
  • Proposition 2.7
  • Proposition 2.8
  • Proof
  • ...and 8 more