The Jordan type of a multiparameter persistence module

Let $\mathscr{P}$ be a poset and $\mathcal{S}$ a sequence of $n$ finite substes of $\mathscr{P}$. The Jordan type of a $\mathscr{P}$-persistence module $M$ at $\mathcal{S}$, denoted by $\mathsf{J}_{\mathcal{S}}(M) \in \mathbb{N}^n$, is defined as the Jordan type of a nilpotent operator $\mathbf{T}_{M, \mathcal{S}}$, which is constructed from $M$ and $\mathcal{S}$. When $n=2$, we recover the notion of multirank previously introduced and studied in [Tho19]. We first prove that the multirank invariants are complete for persistence modules over finite zigzag posets. This proves a conjecture of Thomas in the zigzag case. The nilpotent operator $\mathbf{T}_{M, \mathcal{S}}$ is functorial in $M$. When $\mathscr{P}=\mathbb{Z}^d$ or $\mathbb{R}^d$, this functoriality allows us to define the Jordan filtered rank invariant of $M$ at $\mathscr{S}$. We demonstrate that these invariants are strictly finer than the classical rank invariants. We next prove that for any two $\mathscr{P}$-persistence modules $M$ and $N$, the landscape and erosion distances between their Jordan filtered rank invariants are bounded from above by the interleaving distance between $M$ and $N$.