Cotorsion torsion triples and the representation theory of filtered hierarchical clustering

Ulrich Bauer; Magnus B. Botnan; Steffen Oppermann; Johan Steen

doi:10.1016/j.aim.2020.107171

Cotorsion torsion triples and the representation theory of filtered hierarchical clustering

Ulrich Bauer^*
, Magnus B. Botnan
, Steffen Oppermann
, Johan Steen

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

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Abstract

We give a full classification of representation types of the subcategories of representations of an m×n rectangular grid with monomorphisms (dually, epimorphisms) in one or both directions, which appear naturally in the context of clustering as two-parameter persistent homology in degree zero. We show that these subcategories are equivalent to the category of all representations of a smaller grid, modulo a finite number of indecomposables. This equivalence is constructed from a certain cotorsion torsion triple, which is obtained from a tilting subcategory generated by said indecomposables.

Original language	English
Article number	107171
Pages (from-to)	1-51
Number of pages	51
Journal	Advances in Mathematics
Volume	369
Early online date	27 Apr 2020
DOIs	https://doi.org/10.1016/j.aim.2020.107171
Publication status	Published - 5 Aug 2020

Funding

UB and MB have been supported by the DFG Collaborative Research Center TRR109 Discretization in Geometry and Dynamics . SO has been supported by Norwegian Research Council project 250056 , “Representation theory via subcategories”. JS has been partially supported by Norwegian Research Council project 231000 , “Clusters, combinatorics and computations in algebra”. We thank the anonymous referee for carefully reading the paper and providing valuable comments.

Funders	Funder number
Deutsche Forschungsgemeinschaft
Norges forskningsråd	231000, 250056

Keywords

Hierarchical clustering
Multiparameter persistence
Quiver representation theory
Torsion theory

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Cotorsion torsion triples and the representation theory of filtered hierarchical clusteringFinal published version, 829 KBLicence: Article 25fa Dutch Copyright Act

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KW - Multiparameter persistence

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KW - Torsion theory

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Cotorsion torsion triples and the representation theory of filtered hierarchical clustering

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