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Suppose $G$ is a the cyclic group of $n$ elements and $p$ is a prime not in $n$. $G$ has an action on the cyclotomic field as a $\mathbb{Q}$-vector space $\mathbb{Q}[X] / \langle \Phi_n(X) \rangle$ by multiplication by $X$. This is an irreducible $\mathbb{Q}[G]$-representation. In particular, it acts as automorphisms on $\mathbb{Z}[X]/\langle \Phi_N(X) \rangle$. After reduction modulo $p$, $G$ acts on $\mathbb{F}_p[X]/\langle \Phi_N(X) \rangle$. However, after reduction mod $p$, the representation is no longer irreducible and splits into isomorphic copies the same irreducible representations. This can be shown by seeing how the polynomial $\Phi_N(X)$ splits in the finite field.

One can now consider a finite group $G$ and any prime $p$ not appearing in $|G|$. If we have an irreducible $\mathbb{Q}[G]$-representation $G\rightarrow GL_n(\mathbb{Q})$, after some conjugation, we can assume $G \rightarrow GL_n(\mathbb{Z})$ and then consider the induced $\mathbb{F}_p$-representation. Are there any similar results that generalize the above result of cyclic groups and let us know something about the irreducible components?

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This is touched upon in the following PhD thesis "Integrality of representations of finite groups", see Section 16 on page 84: https://kluedo.ub.uni-kl.de/frontdoor/index/index/year/2016/docId/4408 (your example is Example 16.8.).

You basically have to look at the irreducible $\mathbf C$-representation containing the $\mathbf Q$-representation and play around with classical decomposition theory and Schur indices.

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  • $\begingroup$ The section 16 heavily references the Curtis-Reiner book. Maybe I should read this book first. $\endgroup$ Jun 14, 2019 at 7:53
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    $\begingroup$ Before reading Curtis-Reiner, I'd suggest looking at Part III of Serre's book on the Linear Representations of Finite Groups to see if what you need is there. $\endgroup$
    – anon
    Jun 15, 2019 at 18:06

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