Breaker Chains and Splitting of Interval Submodules in Zigzag Modules
Abstract
Finite-dimensional zigzag modules decompose into interval modules. For an ordinary persistence module, every fully supported interval submodule of a restriction splits as a direct summand, whereas this need not hold for a zigzag module. We study this failure of splitting. A breaker pair is a local configuration of two overlapping interval summands with a backward arrow at the left boundary of the second interval and a forward arrow at the right boundary of the first. A single breaker pair gives a local nonsplitting interval submodule, but it need not control a prescribed larger restriction. We therefore introduce breaker chains. Our main result shows that a restriction V[p,q] contains a nonsplitting submodule isomorphic to I[p,q] if and only if its interval decomposition contains a breaker chain spanning [p,q]. Equivalently, the proper interval summands J for which Hom(I[p,q],J)≠0 have supports covering [p,q]. This cover formulation gives a constructive description of nonsplitting embeddings, a trichotomy for their splitting behavior, and an O(mlog m) algorithm from an oriented barcode with m intervals.
Related
- Semiprime and real submodules of arbitrary modules are intersections of prime and real prime submodules
- Euclidean modules, submodules of their direct sums and strong reduced echelon form
- Splitting of projective modules over some affine algebras
- A minimising principle for interval covers of zigzag persistence modules
- Weak Bruhat interval modules of the 0-Hecke algebras for stable Grothendieck polynomials
BibTeX
@misc{gongopadhyay2026breaker,
title = {{Breaker Chains and Splitting of Interval Submodules in Zigzag Modules}},
author = {Krishnendu Gongopadhyay and Ramveer Singh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14664},
eprint = {2609.14664},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14664},
}
RIS
TY - UNPB
TI - Breaker Chains and Splitting of Interval Submodules in Zigzag Modules
AU - Krishnendu Gongopadhyay
AU - Ramveer Singh
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14664
UR - https://arxiv.org/abs/2609.14664
AB - Finite-dimensional zigzag modules decompose into interval modules. For an ordinary persistence module, every fully supported interval submodule of a restriction splits as a direct summand, whereas this need not hold for a zigzag module. We study this failure of splitting. A breaker pair is a local configuration of two overlapping interval summands with a backward arrow at the left boundary of the second interval and a forward arrow at the right boundary of the first. A single breaker pair gives a local nonsplitting interval submodule, but it need not control a prescribed larger restriction. We therefore introduce breaker chains. Our main result shows that a restriction V[p,q] contains a nonsplitting submodule isomorphic to I[p,q] if and only if its interval decomposition contains a breaker chain spanning [p,q]. Equivalently, the proper interval summands J for which Hom(I[p,q],J)≠0 have supports covering [p,q]. This cover formulation gives a constructive description of nonsplitting embeddings, a trichotomy for their splitting behavior, and an O(mlog m) algorithm from an oriented barcode with m intervals.
ER -