N^d-indexed persistence modules, higher dimensional partitions and rank invariants
Mehdi Nategh, Zhenbo Qin, Shuguang Wang
TL;DR
This work develops a framework for decomposing decomposable $\mathbb N^d$-indexed persistence modules via higher-dimensional partitions and barcodes defined from the extended interiors of Young diagrams. It proves a necessary and sufficient condition, expressed through the partition data, for two barcode-admitting modules to have identical rank invariants; notably, this recovers the classical barcode-rank correspondence when $d=1$ but shows that rank invariants are not complete in higher dimensions. The authors recast $\mathbb N^d$-indexed modules as $\mathbb N^d$-graded $k[t_1,\dots,t_d]$-modules, establish barcode decompositions $M \cong \bigoplus_i \mathbb T_{a^{(i)}} \mathbf k_{\lambda^{(i)}}$, and demonstrate that rank information is captured by per-slice multisets of partition data. They further show that, although the rank invariant constrains the partition content, it does not in general determine the barcode for $d>1$, motivating potential strengthening of conditions to guarantee barcode equality.
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.