### Abstract

We study persistence modules defined on commutative ladders. This class of persistence modules frequently appears in topological data analysis, and the theory and algorithm proposed in this paper can be applied to these practical problems. A new algebraic framework deals with persistence modules as representations on associative algebras and the Auslander–Reiten theory is applied to develop the theoretical and algorithmic foundations. In particular, we prove that the commutative ladders of length less than 5 are representation-finite and explicitly show their Auslander–Reiten quivers. Furthermore, a generalization of persistence diagrams is introduced by using Auslander–Reiten quivers. We provide an algorithm for computing persistence diagrams for the commutative ladders of length 3 by using the structure of Auslander–Reiten quivers.

Original language | English |
---|---|

Pages (from-to) | 100-157 |

Number of pages | 58 |

Journal | Discrete and Computational Geometry |

Volume | 55 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 Jan 1 |

### Keywords

- Auslander–Reiten theory
- Commutative ladder
- Persistence module
- Representation theory
- Topological data analysis

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

## Fingerprint Dive into the research topics of 'Persistence Modules on Commutative Ladders of Finite Type'. Together they form a unique fingerprint.

## Cite this

*Discrete and Computational Geometry*,

*55*(1), 100-157. https://doi.org/10.1007/s00454-015-9746-2