Persistence Modules on Commutative Ladders of Finite Type

TL;DR

This paper develops a new algebraic framework using Auslander-Reiten theory to analyze persistence modules on commutative ladders, providing explicit classifications and algorithms for computing generalized persistence diagrams.

Contribution

It introduces a novel algebraic approach for persistence modules on commutative ladders, proving finiteness for certain lengths and developing algorithms for generalized persistence diagrams.

Findings

Commutative ladders of length less than 5 are representation-finite.

Explicit Auslander-Reiten quivers are constructed for these ladders.

An algorithm for computing persistence diagrams of length 3 ladders is provided.

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.