Grassmannian Persistence Diagrams

We introduce Orthogonal M\"obius Inversion $\mathsf{OI}$, a concept analogous to M\"obius inversion on finite posets, which is applicable to order-preservings functions from a finite poset to the Grassmannian $\mathsf{Gr}(V)$ of an inner product space $V$. This notion critically relies on the inner product structure on $V$ enabling it to capture much finer information than standard integer-valued persistence diagrams. Orthogonal Inversion is a special case of the broader concept of Orthomodular Inversion, where the target space is any orthomodular lattice, which we also identify. We apply Orthogonal Inversion in order to construct a "non-negative" persistence diagram for any given multiparameter filtration $\mathsf{F}$ of a finite simplicial complex $K$, indexed over an arbitrary finite poset $P$. This is done by applying it to the birth-death spaces of $\mathsf{F}$. Analogously to $1$-parameter classical persistence diagrams, these multiparameter Grassmannian persistence diagrams offer straightforward interpretability. Specifically, to a segment $(b, d) \in \mathsf{Seg}(P)$, (1) the Grassmannian persistence diagram canonically assigns a vector subspace of $C_{\rho}^K$ consisting of cycles that are born at $b$ and become boundaries at $d$ and (2) this assignment is exhaustive at the homology level. Finally, we relate our Grassmannian persistence diagrams to the recently introduced notion of M\"obius homology, thus enhancing its interpretability through the lens of our framework.