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Mathematics > Representation Theory

arXiv:2207.03663 (math)
[Submitted on 8 Jul 2022 (v1), last revised 7 Jun 2023 (this version, v3)]

Title:Approximation by interval-decomposables and interval resolutions of persistence modules

Authors:Hideto Asashiba, Emerson G. Escolar, Ken Nakashima, Michio Yoshiwaki
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Abstract:In topological data analysis, two-parameter persistence can be studied using the representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of type A. In a previous work, we defined interval approximations using restrictions to essential vertices of intervals together with Mobius inversion. In this work, we consider homological approximations using interval resolutions, and show that the interval resolution global dimension is finite for finite posets and that it is equal to the maximum of the interval dimensions of the Auslander-Reiten translates of the interval representations. In fact, for the latter equality, we obtained a general formula in the setting of finite-dimensional algebras and resolutions relative to a generator-cogenerator. Furthermore, in the commutative ladder case, by a suitable modification of our interval approximation, we provide a formula linking the two conceptions of approximation.
Subjects: Representation Theory (math.RT); Algebraic Topology (math.AT)
MSC classes: 16G20, 55N31
Cite as: arXiv:2207.03663 [math.RT]
  (or arXiv:2207.03663v3 [math.RT] for this version)
  https://doi.org/10.48550/arXiv.2207.03663
Journal reference: Journal of Pure and Applied Algebra, Vol 227, no.10 (2023): 107397
Related DOI: https://doi.org/10.1016/j.jpaa.2023.107397

Submission history

From: Emerson G. Escolar [view email]
[v1] Fri, 8 Jul 2022 02:56:31 UTC (29 KB)
[v2] Wed, 10 Aug 2022 03:15:41 UTC (27 KB)
[v3] Wed, 7 Jun 2023 03:12:53 UTC (35 KB)
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