Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules

Abstract

We prove that pointwise finite-dimensional S^1 persistence modules over an arbitrary field decompose uniquely, up to isomorphism, into the direct sum of a bar code and finitely-many Jordan cells. These persistence modules have also been called angle-valued or circular persistence modules. We allow either a cyclic order or partial order on S^1 and do not have additional finiteness requirements on the modules. We also show that a pointwise finite-dimensional S^1 persistence module is indecomposable if and only if it is a bar or Jordan cell (a string or a band module, respectively, in representation theory). Along the way we classify the isomorphism classes of such indecomposable modules.

16 pages. 2 figures. v3: Typos fixed and DOI added to a reference. v2: Minor changes to abstract. Expanded discussion with additional references, particularly in the introduction. Added results about isoclasses and endomorphism rings of indecomposable modules

References in corpus (4)

• Hierarchical structures of amorphous solids characterized by persistent homology
• Topological Data Analysis of Biological Aggregation Models
• Persistent homology and Floer-Novikov theory
• Fock space representation of the circle quantum group

Cited by in corpus (1)

• Continuous Nakayama Representations