In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a wild problem. One direction is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more natural algebraic definition, and study the relationships between pre-interval, interval, and thin indecomposable representations. We show that over the “equioriented” commutative 2D grid, these concepts are equivalent. Moreover, we provide a criterion for determining whether or not an nD persistence module is interval/pre-interval/thin-decomposable without having to explicitly compute decompositions. For 2D persistence modules, we provide an algorithm for determining interval-decomposability, together with a worst-case complexity analysis that uses the total number of intervals in an equioriented commutative 2D grid. We also propose several heuristics to speed up the computation.

While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and complete descriptor analogous to the persistence diagrams of the former. We propose a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size. On top of providing a constructive proof of representation infinite type, we also provide realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies.