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Let $G$ be a finite group and let $\rho\colon G\to \mathrm{GL}_n(\mathbb{Q})$ be a representation of $G$.

How does one go about classifying the $\mathbb{Z}$-forms of $\rho$?

For example: here it is claimed that the $2$-dimensional representation of $\mathrm{Sym}(3)$ has $2$ distinct $\mathbb{Z}$-forms --- how is this proven? What happens for the standard representation of $\mathrm{Sym}(4)$?

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  • $\begingroup$ For 2-dimensional things are particularly simple because one easily understands finite subgroups of $\mathrm{GL}_2(\mathbf{Z})$ up to conjugation. $\endgroup$
    – YCor
    Commented Jun 29, 2023 at 22:45
  • $\begingroup$ You can distinguish the two $\mathbb{Z}$-forms by reducing mod $3$ and looking for fixed points. $\endgroup$
    – Dave Benson
    Commented Jul 5, 2023 at 8:39
  • $\begingroup$ @DaveBenson thanks! Does something similar happen for Sym(4)? What about Sym(n)? I guess classifying Z-forms is less procedural and more ad-hoc? $\endgroup$
    – Sam
    Commented Aug 7, 2023 at 15:27

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I recommend looking at Chapters 3 and 4 of the new Curtis and Reiner, "Methods of Representation Theory I" (...when I say new, I mean 1981! Am I showing my age?) as well as Reiner's "Maximal Orders" (1974). The subject is subtle, but there are some good results.

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