Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866908411618131968 |
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| author | Hanson, Eric J. Rock, Job D. |
| author_facet | Hanson, Eric J. Rock, Job D. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2006_13793 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules Hanson, Eric J. Rock, Job D. Representation Theory Algebraic Topology 16G20, 55N31 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. |
| title | Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules |
| topic | Representation Theory Algebraic Topology 16G20, 55N31 |
| url | https://arxiv.org/abs/2006.13793 |