$\mathbb{Z}$ is the simplest example of a group with indecomposable representations which are not irreducible. Over $\mathbb{C}$, the isomorphism classes of $n$-dimensional representations of $\mathbb{Z}$ are precisely the conjugacy classes of elements of $GL_n(\mathbb{C})$; in particular the indecomposable ones are given by Jordan blocks. The representation corresponding to a Jordan block of size $n$ with diagonal entries $\lambda$ has the same character as, but is not isomorphic to, the representation corresponding to a diagonal matrix with entries $\lambda$. What is an abstract way to describe this relationship that does not refer to characters? (I am mostly interested in how to describe the relationship between an indecomposable representation and a sum of one-dimensional representations with the same character.)
Motivation
A natural way to study an (associative, unital) algebra $A$ over $\mathbb{C}$ (to fix ideas) is to study the category $\text{Rep}(A)$ of, say, finite-dimensional representations of $A$. However, if $A$ happens to be commutative and Noetherian, then we do something different: we privilege the one-dimensional representations and call them points, and then we analyze higher-dimensional representations as certain structures on the points. What abstract relationship, from the representation-theoretic perspective, between the one-dimensional and higher-dimensional representations lets us do this?
