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In his answer to my question here, Victor Protsak quoted the following result:

Let $C_2$ be a finite cyclic group of order $2$. Then every $\mathbb{Z}[C_2]$ structure on $\mathbb{Z}^n$ is isomorphic to a direct sum of representations of the following types:

  1. The trivial representation
  2. The sign representation
  3. The representation on $\mathbb{Z}^2$ that permutes the two factors.

This strikes me as a very beautiful result! I know a lot of nice sources for representation theory over various kinds of fields, but a bit of searching does not turn up any books or surveys on representation theory over the integers. Does anyone have anything they recommend?

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See Curtis–Reiner's textbook on the Representation Theory of Finite Groups and Associative Algebras (MR 144979), Theorem 74.3, page 507, and especially the introduction starting on page 493.

The result for cyclic groups of prime order, and for order 4 was originally done in:

  • Diederichsen, Fritz-Erdmann. "Über die Ausreduktion ganzzahliger Gruppendarstellungen bei arithmetischer Äquivalenz" Abh. Math. Sem. Hansischen Univ. 13, (1940). 357–412. MR2133.

However, Reiner has written quite a few nice papers on similar subjects. One of his earlier ones is on the same topic:

  • Reiner, Irving. "Integral representations of cyclic groups of prime order." Proc. Amer. Math. Soc. 8 (1957), 142–146. MR83493 doi:10.2307/2032829

One can also consult texts on things called "crystallographic groups", "space groups", and "p-adic space groups". Plesken has written several nice books using this sort of thing. These give infinite families of nicely related finite groups and of course help crystallographers.

Be careful to distinguish these sorts of representations from ZG-modules. ZG-modules are basically incomprehensible, so instead lots of people focus on ZG-lattices, where the underlying module is projective. This means the idea of using matrices still makes some sense. There is a lot of literature on modules over group rings over nice rings (like Z or Dedekind domains), but a fair amount of it is not applicable to questions about GL(n,Z).

Roughly speaking, even for G=1, ZG modules are too difficult to understand, and adding a G just makes it worse. Another common tack is to look at $\hat {\mathbb{Z}}_p$ modules, p-adic modules. Again the results are nicest for lattices, but things do not get so bad near so fast there. Reiner's Maximal Orders textbook describes some of the beautiful and well-behaved things you can see there.

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    $\begingroup$ I don't know whether "ZG-modules are basically incomprehensible" is a faithful representation. The usual motivations to study representations come either from matrix groups over a field or from lattices, so, naturally, these two cases have been most developed. Also, finitely generated Z-modules (aka abelian groups) are NOT too difficult to understand. On the contrary, there is a beautiful complete theory which was the source of many developments in group theory and algebra in general. $\endgroup$
    – Victor Protsak
    Commented Jun 5, 2010 at 23:50
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    $\begingroup$ +1 -> Victor for gratuitous use of "faithful representation". $\endgroup$
    – Allen Knutson
    Commented Jun 6, 2010 at 1:30
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    $\begingroup$ Diederichsen's treatment of $\mathbb Z/4$ is actually incorrect, see ams.org/mathscinet-getitem?mr=124418 $\endgroup$
    – Rasmus
    Commented Jun 1, 2011 at 12:21

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