Interval Multiplicities of Persistence Modules

Abstract: For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ in $\mathbf{P}$, we give a formula of the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of the ranks of matrices consisting of structure linear maps of the module $M$, which gives a generalization of the corresponding formula for 1-dimensional persistence modules. This makes it possible to compute the maximal interval-decomposable direct summand of $M$, which gives us a way to decide whether $M$ is interval-decomposable or not. Moreover, the formula tells us which morphisms of $\mathbf{P}$ are essential to compute the multiplicity $d_M(V_I)$. This suggests us some order-preserving map $\zeta \colon Z \to \mathbf{P}$ such that the induced restriction functor $R \colon \operatorname{mod} \mathbf{P} \to \operatorname{mod} Z$ has the property that the multiplicity $d:= d_{R(M)}(R(V_I))$ is equal to $d_M(V_I)$. In this case, we say that $\zeta$ essentially covers $I$. If $Z$ can be taken as a poset of Dynkin type $\mathbb{A}$, also known as a zigzag poset, then the calculation of the multiplicity $d$ can be done more efficiently, starting from the filtration level of topological spaces. Thus this even makes it unnecessary to compute the structure linear maps of $M$.