Grassmannian Persistence Diagrams

We introduce Orthogonal Möbius Inversion, a concept analogous to Möbius inversion on finite posets, applicable to order-preserving functions from a finite poset to the Grassmannian $\mathrm{Gr}(V)$ of an inner product space $V$. This notion relies critically on the inner product structure on $V$, enabling it to capture finer information than standard integer-valued persistence diagrams.

Orthogonal inversion is a special case of the broader concept of orthomodular inversion, in which the target is an arbitrary orthomodular lattice. We apply orthogonal inversion to construct a “nonnegative” persistence diagram for any given multiparameter filtration $F$ of a finite simplicial complex $K$, indexed over an arbitrary finite poset $P$, by applying it to the birth–death spaces of $F$.

Analogously to classical one-parameter persistence diagrams, these multiparameter Grassmannian persistence diagrams admit a straightforward interpretation. Specifically, for each segment $(b,d)\in \mathrm{Seg}(P)$:

1. the Grassmannian persistence diagram canonically assigns a vector subspace of degree-$*$ cycles in $K$ that are born at $b$ and become boundaries at $d$, and

2. this assignment is exhaustive at the homology level.