# Mod $p$ of irreducible representation is irreducible for large $p$?

Let $$G$$ be a finite group, and a homomophism $$f: G \rightarrow GL_n(\mathbb Z)$$. Assume $$f \otimes \mathbb Q: G \rightarrow GL_n(\mathbb Z) \hookrightarrow GL_n(\mathbb Q)$$ is an irreducible $$\mathbb Q$$-representation of $$G$$, must $$f \otimes \mathbb F_p: G \rightarrow GL_n(\mathbb Z) \rightarrow GL_n(\mathbb F_p)$$ is an irreducible $$\mathbb F_p$$-representation of $$G$$ for large enough $$p$$?

If $$f \otimes \mathbb Q$$ is absolute irreducible, one can show this by character theory.

## 1 Answer

No. For instance, let $$G$$ be cyclic of order $$4$$ and let $$f:G\to GL_2(\mathbb{Z})$$ send a generator to $$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$. Then for any field $$K$$, $$f\otimes K$$ is reducible iff this matrix has an eigenvalue in $$K$$, i.e. iff $$-1$$ has a square root in $$K$$. So, $$f\otimes\mathbb{Q}$$ is irreducible, but $$f\otimes\mathbb{F}_p$$ is reducible for any prime $$p$$ that is $$1$$ mod $$4$$, and there are infinitely many such $$p$$.