Grassmannian Persistence Diagrams: Special Properties in the 1-Parameter Setting

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In this paper, we explore the discriminative power of Grassmannian persistence diagrams of $1$-parameter filtrations, examine their relationships with other related constructions, and study their computational aspects. Grassmannian persistence diagrams are defined through Orthogonal Inversion, a notion analogous to Möbius inversion. We focus on the behavior of this inversion for the poset of segments of a linear poset. We demonstrate how Grassmannian persistence diagrams of $1$-parameter filtrations are connected to persistent Laplacians via a variant of orthogonal inversion tailored for the reverse-inclusion order on the poset of segments. Additionally, we establish an explicit isomorphism between Grassmannian persistence diagrams and Harmonic Barcodes via a projection. Finally, we show that degree-$0$ Grassmannian persistence diagrams are equivalent to treegrams, a generalization of dendrograms. Consequently, we conclude that finite ultrametric spaces can be recovered from the degree-$0$ Grassmannian persistence diagram of their Vietoris-Rips filtrations.