A persitence module is a functor $F:\mathbb{N} \rightarrow \mathbb{A}$ where $\mathbb{N}$ is the category of natural numbers with a partial order and $\mathbb{A}$ is some abelian category. There is a theorem by Crawley-Bovey that gives conditions on when a persitence module decomposes into interval modules. In the case that the partial order is the standard order on $\mathbb{N}$ and $\mathbb{A}$ is the category of vector spaces $V$ over a finite field $F$ with linear maps as morphisms. Then an interval module looks like (ignoring the index):
$$ \ldots \rightarrow 0\rightarrow 0 \rightarrow V \rightarrow V \rightarrow \ldots \rightarrow V\rightarrow 0 \rightarrow 0 \rightarrow \ldots $$
I was wondering wether anyone had a list of examples of persistence modules that are not decomposable into interval modules. It would be particularly interesting if there are examples that arise from studying a data set or topological persitence.
Here are two examples of nondecomposable persistence modules.
The first one is due to Cary Webb (1985). Let $\mathbb{V}$ be the persistence module indexed over the nonpositive integers (or over the nonnegative integers with the reverse ordering), where $$\begin{align*} V_0 &= \{\text{sequences $(x_1, x_2, \ldots)$ of scalars}\} \\ V_{-n} &= \{\text{sequences with $x_1 = x_2 = \cdots = x_n = 0$}\}, \text{if } n > 0. \end{align*}$$ The maps are the canonical inclusion maps. Suppose $\mathbb{V}$ had an interval decomposition. Since all the maps are inclusions, the intervals occuring in the decomposition must have the form $[-n, 0]$ or $(-\infty, 0]$. The multiplicity of the interval module corresponding to $(-\infty, 0]$ is zero, since the only element in this module would be the zero sequence. The multiplicity of $[-n, 0]$ is $\dim(V_{-n} / V_{-n-1}) = 1$. Therefore, if $\mathbb{V}$ had an interval decomposition it must be $\bigoplus_{n \geq 0} [-n, 0]$. This implies that $\dim V_0$ is countable, but actually it is uncountable!
This second example is due to Michael Lesnick. Let $\mathbb{V}$ be the persistence module indexed over the nonpositive integers, where $$\begin{align*} V_0 &= k \\ V_{-1} &= \{\text{eventually zero sequences $(x_1, x_2, \ldots)$}\} \\ V_{-n} &= \{\text{such sequences with $x_1 = \cdots = x_{n-1} = 0$}\}, \, \text{if } n > 1. \end{align*}$$ For $-m \leq -n < 0$, the map $V_{-m} \to V_{-n}$ is the canonical map. If $-n = 0$, the map is the map that takes the sum of the (finitely many nonzero) entries of the sequence. If $\mathbb{V}$ had an interval decomposition, there is exactly one module of the form $[-n, 0]$ or $(-\infty, 0]$ in the decomposition since $\dim V_0 = 1$. In fact, it must be $(-\infty ,0]$ since $V_{-n} \to V_0$ is nontrivial for all $n$. Now, what is the basis element in this module at $V_{-1}$? It is the nontrivial image of an element in $V_{-n}$ with nonzero sum for all $n$, so it cannot be eventually zero, contradiction.
These examples are all rather contrived. For persistence modules arising from "data" in the usual way, which are pointwise finite dimensional, failure of interval decomposability can never happen, due to the result of Crawley-Boevey you allude to.