Persistence module and Schubert calculus

Abstract

A multiplication on persistence diagrams is introduced by means of Schubert calculus. The key observation behind this multiplication comes from the fact that the representation space of persistence modules has the structure of the Schubert decomposition of a flag. In particular, isomorphism classes of persistence modules correspond to Schubert cells, thereby the Schubert calculus naturally defines a multiplication on persistence diagrams. The meaning of the multiplication on persistence diagrams is carried over from that on Schubert calculus, i.e., algebro-geometric intersections of varieties of persistence modules.