Tensor, Symmetric, Exterior, and Other Powers of Persistence Modules

TL;DR

This paper reformulates persistent (co)homology using algebraic modules over polynomial rings and introduces formulas for tensor, symmetric, and exterior powers of persistence modules, extending to group actions.

Contribution

It provides an algebraic reformulation of persistent (co)homology and derives formulas for tensor, symmetric, and exterior powers of persistence modules, including group actions.

Findings

Formulated persistent (co)homology as chain complexes of graded modules.

Derived formulas for tensor, symmetric, and exterior powers of persistence modules.

Extended the framework to cyclic and dihedral powers, and quotients by group actions.

Abstract

We reformulate the persistent (co)homology of simplicial filtrations, viewed from a more algebraic setting, namely as the (co)homology of a chain complex of graded modules over polynomial ring $K[t]$. We also define persistent (co)homology of groups, associative algebras, Lie algebras, etc. \par Then we obtain formulas for tensor powers $T^n(M),S^n(M),\Lambda^{\!n}(M)$ where $M$ is a persistence module. We discuss the cyclic and dihedral powers of persistence modules, and more generally quotients of $T^n(M)$ by a group action.