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Classifying Nakayama algebras with a braid group action on $τ$-exceptional sequences

Maximilian Kaipel, Håvard Utne Terland

TL;DR

This work settles when mutation of $ au$-exceptional sequences over Nakayama algebras yields a braid group action. By melding $ au$-tilting theory, the mutation theory of $ au$-exceptional sequences, and Jasso/Drozd–Kirichenko reductions, the authors prove that braid relations hold iff the algebra is hereditary or every indecomposable projective has length at least $|rac{|\\Lambda|}$; in particular, they verify braid relations for the $C_n$-type Nakayama algebras and their quotients. They establish a precise bijection framework to transfer mutation behavior across algebras with large projective lengths, and they provide explicit counterexamples demonstrating the necessity of the stated condition. The results connect classical exceptional-sequence mutations in hereditary settings with the more general $ au$-tilting landscape, offering a complete classification of non-hereditary Nakayama algebras admitting braid-group mutations.

Abstract

We characterise those basic and connected Nakayama algebras $Λ$ for which the mutation of $τ$-exceptional sequences respects the braid group relations. We show that this is the case if and only if $Λ$ is hereditary or all indecomposable projective $Λ$-modules have length at least $|Λ|$.

Classifying Nakayama algebras with a braid group action on $τ$-exceptional sequences

TL;DR

This work settles when mutation of -exceptional sequences over Nakayama algebras yields a braid group action. By melding -tilting theory, the mutation theory of -exceptional sequences, and Jasso/Drozd–Kirichenko reductions, the authors prove that braid relations hold iff the algebra is hereditary or every indecomposable projective has length at least ; in particular, they verify braid relations for the -type Nakayama algebras and their quotients. They establish a precise bijection framework to transfer mutation behavior across algebras with large projective lengths, and they provide explicit counterexamples demonstrating the necessity of the stated condition. The results connect classical exceptional-sequence mutations in hereditary settings with the more general -tilting landscape, offering a complete classification of non-hereditary Nakayama algebras admitting braid-group mutations.

Abstract

We characterise those basic and connected Nakayama algebras for which the mutation of -exceptional sequences respects the braid group relations. We show that this is the case if and only if is hereditary or all indecomposable projective -modules have length at least .

Paper Structure

This paper contains 9 sections, 40 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. $\tau$-tilting theory
  4. Mutation of $\tau$-exceptional sequences
  5. Mutation of $\tau$-exceptional sequences over Nakayama algebras
  6. Mutation of $\tau$-exceptional sequences as group actions
  7. Braid group relations for Nakayama algebras of the form $C_n$
  8. Braid group relations for Nakayama algebras having $C_n$ as a quotient
  9. Counterexamples for the remaining non-hereditary cases

Key Result

Theorem 1.1

Let $C_n$ be the Nakayama algebra given as a quotient of the $n$-cycle path algebra over a field $k$, modulo the ideal generated by paths of length $n$. Then mutation of $\tau$-exceptional sequences over $C_n$ respects the braid group relations.

Figures (1)

  • Figure 1: The AR-quiver of $C_6$ with the subcategory $J(M^3_3)$ of $\mathsf{mod} C_6$ highlighted in blue and green. Notice that the modules with top $S(1)$ are drawn twice. The hereditary component of $J(M^3_3)$ is illustrated by the green boundary and the cyclic component is illustrated by the blue boundary.

Theorems & Definitions (87)

  • Theorem 1.1: \ref{['thm:braid_group_relations_satisfied_on_cn']}
  • Theorem 1.2: \ref{['thm:longerlengths']}, reformulated
  • Theorem 1.3: \ref{['thm:notethm1']}
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 77 more