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$\mathbb{Z}$ 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?

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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. –  Mariano Suárez-Alvarez Apr 5 '10 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. –  moonface Apr 6 '10 at 4:15
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1 Answer 1

If you don't mind noncommutative finite dimensional examples, then Külshammer's Lectures on Block Theory is one (of several) books that describe generalizations of points to larger classes. His book is very simple and basically identifies idempotents with points (I think commutative people call this "dimension 0"). If this is acceptable, then the book by Thévenaz on G-Algebras is likely to be what you are looking for.

If you want to more smoothly transition between characters and representation rings, then Benson's Modular Representation Theory LNM1081 DOI:10.1007/3-540-38940-7 describes a nice framework to work in called species.

Is it ok just to say that your indecomposable representation has the following composition factors? In the finite dimensional case that is all you are saying, and it doesn't seem too different in general.

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Yes, it sounds like Mariano's suggestion of semisimplification is exactly what I'm talking about. Thanks! –  Qiaochu Yuan Apr 5 '10 at 21:52
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