# What is the abstract relationship between an indecomposable representation and a sum of irreducible representations with the same character?

$\mathbb{Z}$ 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?

• For $\mathbb Z$, the «representation corresponding to a diagonal matrix with entries $\lambda$» can be described as the semisimplification of the original module, the direct sum of the subquotients appearing in a composition series. You can do that for all groups and modules. Apr 5, 2010 at 21:19
• Take the conjugacy class of your (not necessarily semisimple) matrix inside $GL(n, \mathbb{C})$, and take its Zariski closure (or its closure in the usual topology). This contains a unique semisimple conjugacy class. Apr 6, 2010 at 4:15