Refinement of interval approximations for fully commutative quivers

A central challenge in the theory of multiparameter persistence modules lies in defining effective descriptors for representations of infinite or wild type. In this work, we propose a novel framework for analyzing interval approximations of fully commutative quivers, which offers a tunable trade-off between approximation resolution and computational complexity. Our approach is evaluated on commutative ladder modules of both finite and infinite type. For finite-type cases, we establish an efficient method for computing indecomposable decompositions using solely one-parameter persistent homology. For infinite-type cases, we introduce a new invariant that captures persistence in the second parameter by connecting standard persistence diagrams through interval approximations. Furthermore, we present several models for constructing commutative ladder filtrations, providing new insights into the behavior of random filtrations and demonstrating the utility of our framework in topological analysis of material structures.