N^d-indexed persistence modules, higher dimensional partitions and rank invariants
Algebraic Topology
2025-10-29 v1 Algebraic Geometry
Combinatorics
Abstract
We study decomposable N^d-indexed persistence modules via higher dimensional partitions. Their barcodes are defined in terms of the extended interior of the corresponding Young diagrams. For two decomposable N^d-indexed persistence modules, we present a necessary and sufficient condition, in terms of the partitions, for their rank invariants to be the same. This generalizes the well-known fact that for an N-indexed persistence module, its barcode and its rank invariant determine each other, i.e., the rank invariant is a complete invariant.
Cite
@article{arxiv.2510.23811,
title = {N^d-indexed persistence modules, higher dimensional partitions and rank invariants},
author = {Mehdi Nategh and Zhenbo Qin and Shuguang Wang},
journal= {arXiv preprint arXiv:2510.23811},
year = {2025}
}
Comments
To appear in AIMS Mathematics Special Issue "Recent Advances in Algebraic Topology and Applications"