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The representation theory of finite groups over the fields $\mathbb{C}$ or $\mathbb{Q}$ is clear. I am wondering what happens if we consider something similar over the group $\mathbb{Z}$. To be more precise, let $G$ be a finite subgroup of $\operatorname{GL}_n(\mathbb{Z})$, and consider the action of $G$ on $\mathbb{Z}^n$.

Question: How to describe $G$-invariant subgroups of $\mathbb{Z}^n$?

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    $\begingroup$ I'm not sure what kind of answer you want for your specific question, but more generally you are looking at "integral representation theory", and it is very hard. You can find some stuff about it in Chapter XI of Curtis and Reiner, but there is nothing approaching the nice picture we have over fields of characteristic 0. $\endgroup$ Commented Jun 5, 2024 at 18:05
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    $\begingroup$ I also recommend Irv Reiner's "Maximal Orders" for integral representation theory. $\endgroup$ Commented Jun 6, 2024 at 9:43

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