N^d-indexed persistence modules, higher dimensional partitions and rank invariants

TL;DR

This paper extends the concept of barcodes and rank invariants from one-dimensional persistence modules to higher dimensions using partitions and Young diagrams, providing conditions for their equivalence.

Contribution

It introduces a higher-dimensional framework for persistence modules, defining barcodes via Young diagrams and establishing criteria for rank invariants to fully determine modules.

Findings

Characterization of barcodes using extended Young diagrams

Necessary and sufficient conditions for rank invariants to match

Generalization of the one-dimensional barcode-rank invariant relationship

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.