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.
Keywords
Cite
@article{arxiv.2006.13793,
title = {Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules},
author = {Eric J. Hanson and Job D. Rock},
journal= {arXiv preprint arXiv:2006.13793},
year = {2025}
}
Comments
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