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Mutating ordered $τ$-rigid modules with applications to Nakayama algebras

Aslak B. Buan, Maximilian Kaipel, Håvard U. Terland

TL;DR

This work develops a mutation framework for $\tau$-exceptional sequences by translating mutation into operations on TF-ordered $\tau$-rigid modules, via the $V$- and $E$-maps. It provides explicit mutation rules and completions in the Nakayama setting, including Bongartz and co-Bongartz constructions, and shows that mutation is transitive for Nakayama algebras by combining direct Nakayama-case formulas with $\tau$-tilting reduction. The results yield a concrete combinatorial description of mutations, enable detailed analysis of the mutation graph, and connect to geometric disk models to illustrate the mutation landscape. These contributions advance understanding of mutation phenomena beyond hereditary algebras and illuminate structural properties of $\tau$-tilting theory in the Nakayama context, with potential implications for broader classes of algebras.

Abstract

A mutation operation for $τ$-exceptional sequences of modules over any finite-dimensional algebra was recently introduced, generalising the mutation for exceptional sequences of modules over hereditary algebras. We interpret this mutation in terms of TF-ordered $τ$-rigid modules, which are in bijection with $τ$-exceptional sequences. As an application we show that the mutation is transitive for Nakayama algebras, by providing an explicit combinatorial description of mutation over this class of algebras.

Mutating ordered $τ$-rigid modules with applications to Nakayama algebras

TL;DR

This work develops a mutation framework for -exceptional sequences by translating mutation into operations on TF-ordered -rigid modules, via the - and -maps. It provides explicit mutation rules and completions in the Nakayama setting, including Bongartz and co-Bongartz constructions, and shows that mutation is transitive for Nakayama algebras by combining direct Nakayama-case formulas with -tilting reduction. The results yield a concrete combinatorial description of mutations, enable detailed analysis of the mutation graph, and connect to geometric disk models to illustrate the mutation landscape. These contributions advance understanding of mutation phenomena beyond hereditary algebras and illuminate structural properties of -tilting theory in the Nakayama context, with potential implications for broader classes of algebras.

Abstract

A mutation operation for -exceptional sequences of modules over any finite-dimensional algebra was recently introduced, generalising the mutation for exceptional sequences of modules over hereditary algebras. We interpret this mutation in terms of TF-ordered -rigid modules, which are in bijection with -exceptional sequences. As an application we show that the mutation is transitive for Nakayama algebras, by providing an explicit combinatorial description of mutation over this class of algebras.
Paper Structure (15 sections, 57 theorems, 123 equations, 6 figures)

This paper contains 15 sections, 57 theorems, 123 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. $\tau$-tilting theory
  4. $\tau$-exceptional sequences and TF-ordered $\tau$-rigid modules
  5. Mutation of $\tau$-exceptional sequences and corresponding TF-ordered modules
  6. The $V$ and $E$-maps
  7. Some operations on TF-ordered $\tau$-rigid modules
  8. Mutation of $\tau$-exceptional sequences of length $2$
  9. Mutation of $\tau$-exceptional sequences of length $>2$
  10. Mutation of TF-ordered modules for Nakayama algebras
  11. Bongartz completions and $\tau$-tilting reduction
  12. Left regular mutation
  13. Left irregular mutation
  14. Transitivity of mutation for Nakayama algebras
  15. Examples and illustrations

Key Result

Theorem 1.1

(prop:TFadmissiblebehaviour) Let $B \oplus C$ be a TF-ordered $\tau$-rigid module. If the corresponding $\tau$-exceptional pair is left regular, see BHM2024 then the left mutation described in BHM2024 corresponds to the following mutation of TF-ordered $\tau$-rigid modules: where we write $C[1]$ to mean the $\tau$-rigid pair $(0,C)$.

Figures (6)

  • Figure 1: Bongartz completion (left) and co-Bongartz completion (right) of an indecomposable $\tau$-rigid $M \cong M_{s,t}$ on Adachi's disk model
  • Figure 2: The module $\widehat{B}$ of \ref{['thm:Nakayamacase4']} (left) and the quiver (in orange) of $\mathop{\mathrm{\mathrm{End}}}\nolimits(\mathrm{B}(M))/\langle M \rangle$ embedded into Adachi's disk model (right)
  • Figure 3: The $\overline{\varphi}$-orbit of a TF-ordered $\tau$-rigid module $B \oplus C$ illustrating three cases of \ref{['thm:introNakmutation']}. The left direct summand corresponds to the orange arc and the right direct summand corresponds to the blue arc.
  • Figure 4: The $\overline{\varphi}$-orbit of a TF-ordered $\tau$-rigid module $B \oplus C$ illustrating the remaining three cases of \ref{['thm:introNakmutation']}. The left direct summand corresponds to the orange arc and the right direct summand corresponds to the blue arc.
  • Figure 5: A $\varphi$-orbit of length 4, which cannot come from applying BHM2024 to any functorially finite wide subcategory of rank 2. The left direct summand corresponds to the orange arc and the right direct summands corresponds to the blue arc.
  • ...and 1 more figures

Theorems & Definitions (136)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Proposition 2.6
  • ...and 126 more