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

2025/10/27 by Nategh, Mehdi, Qin, Zhenbo, Wang, Shuguang
#05A17 #14C05 #55N31 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.23811

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.

Citations

• CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizations
• A Schauder Basis for Multiparameter Persistence
• Interval Multiplicities of Persistence Modules
• Multiparameter Persistence Landscapes

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