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

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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 languageEnglish
Article number107171
Pages (from-to)1-51
Number of pages51
JournalAdvances in Mathematics
Volume369
Early online date27 Apr 2020
DOIs
Publication statusPublished - 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.

FundersFunder number
Deutsche Forschungsgemeinschaft
Norges forskningsråd231000, 250056

    Keywords

    • Hierarchical clustering
    • Multiparameter persistence
    • Quiver representation theory
    • Torsion theory

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