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Matrix method for persistence modules on commutative ladders of finite type

Authors
  • Asashiba, Hideto1
  • Escolar, Emerson G.2
  • Hiraoka, Yasuaki2, 3
  • Takeuchi, Hiroshi4, 5
  • 1 Shizuoka University, Department of Mathematics, Faculty of Science, Shizuoka, Japan , Shizuoka (Japan)
  • 2 Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan , Tokyo (Japan)
  • 3 Kyoto University, Kyoto University Institute for Advanced Study, Kyoto, Japan , Kyoto (Japan)
  • 4 Tohoku University, Mathematics Department, Graduate School of Science, Sendai, Japan , Sendai (Japan)
  • 5 Japan Society for the Promotion of Science, Tokyo, Japan , Tokyo (Japan)
Type
Published Article
Journal
Japan Journal of Industrial and Applied Mathematics
Publisher
Springer Japan
Publication Date
Sep 24, 2018
Volume
36
Issue
1
Pages
97–130
Identifiers
DOI: 10.1007/s13160-018-0331-y
Source
Springer Nature
Keywords
License
Yellow

Abstract

The theory of persistence modules on the commutative ladders CLn(τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CL_n(\tau )$$\end{document} provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view a persistence module M on CLn(τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CL_n(\tau )$$\end{document} as a morphism between zigzag modules, which can be expressed in a block matrix form. For the representation finite case (n≤4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\le 4$$\end{document}), we provide an algorithm that uses certain permissible row and column operations to compute a normal form of the block matrix. In this form an indecomposable decomposition of M, and thus its persistence diagram, is obtained.

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