Decomposition of pointwise finite-dimensional 𝕊1 persistence modules

Abstract

We prove that, over an arbitrary field, pointwise finite-dimensional persistence modules indexed by [Formula: see text] decompose uniquely, up to isomorphism, into the direct sum of a bar code and finitely-many Jordan cells. In the language of representation theory, this is a direct sum of string modules and band modules. Persistence modules indexed on [Formula: see text] have also been called angle-valued or circular persistence modules. We allow either a cyclic order or partial order on [Formula: see text] and do not have additional finiteness requirements on the modules. We also show that a pointwise finite-dimensional [Formula: see text] persistence module is indecomposable if and only if it is a bar or Jordan cell. Along the way we classify the isomorphism classes of such indecomposable modules.