Tilting mutation of gentle algebras and its combinatorial description
Abstract
Tilting mutation provides a natural way to construct derived-equivalent algebras by replacing an indecomposable summand of a tilting object. Holm, Schröer and Zimmermann~\cite{HSZ01} (see also the appendix of~\cite{AA08}) have described combinatorial transformations of the quiver and relations of a gentle algebra that yield derived equivalences via tilting complexes. In this paper, we develop a self-contained module-theoretic framework for tilting mutation via generalized BB-tilting modules. For a gentle algebra A=k Q/⟨ I⟩ and a vertex k∈ Q0, we first give a necessary and sufficient condition, expressed in terms of the arrows and relations incident with k, for the corresponding minimal left approximation of P(k) to yield a tilting mutation. This criterion applies uniformly to vertices with or without loops. When the mutation exists, we determine the irreducible morphisms between the indecomposable summands of the mutated tilting module and use them to construct explicitly the Gabriel quiver and defining relations of its endomorphism algebra. In particular, we also provide a combinatorial mutation (Q,I)↦(Q',I') of gentle pairs such that μk+(A)≅ k Q'/⟨ I'⟩, so the resulting algebra is again gentle and derived equivalent to A. Finally, we formulate the dual cotilting mutation and its combinatorial description via opposite gentle pairs.
Related
- The Hochschild (co)homology of gentle algebras
- Transferred L∞-structures and a classification of binary brackets for finite-dimensional skew-gentle algebras
- Extriangulated ideal quotients, with applications to cluster theory and gentle algebras
- Support τ-tilting modules over Morita context algebras: A bilateral approximation approach
- Combinatorial description of the mirabolic quantum Schur-Weyl duality
BibTeX
@misc{deng2026tilting,
title = {{Tilting mutation of gentle algebras and its combinatorial description}},
author = {Difan Deng and Shengfei Geng and Pin Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39536},
eprint = {2609.39536},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39536},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Tilting mutation of gentle algebras and its combinatorial description
AU - Difan Deng
AU - Shengfei Geng
AU - Pin Liu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.39536
UR - https://arxiv.org/abs/2609.39536
AB - Tilting mutation provides a natural way to construct derived-equivalent algebras by replacing an indecomposable summand of a tilting object. Holm, Schröer and Zimmermann~\cite{HSZ01} (see also the appendix of~\cite{AA08}) have described combinatorial transformations of the quiver and relations of a gentle algebra that yield derived equivalences via tilting complexes. In this paper, we develop a self-contained module-theoretic framework for tilting mutation via generalized BB-tilting modules. For a gentle algebra A=k Q/⟨ I⟩ and a vertex k∈ Q_0, we first give a necessary and sufficient condition, expressed in terms of the arrows and relations incident with k, for the corresponding minimal left approximation of P(k) to yield a tilting mutation. This criterion applies uniformly to vertices with or without loops. When the mutation exists, we determine the irreducible morphisms between the indecomposable summands of the mutated tilting module and use them to construct explicitly the Gabriel quiver and defining relations of its endomorphism algebra. In particular, we also provide a combinatorial mutation (Q,I)↦(Q',I') of gentle pairs such that μ_k^+(A)≅ k Q'/⟨ I'⟩, so the resulting algebra is again gentle and derived equivalent to A. Finally, we formulate the dual cotilting mutation and its combinatorial description via opposite gentle pairs.
ER -