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Let $G$ be a group, and $\rho: G \rightarrow \textrm{GL}(V)$ is an irreducible abstract representation of $G$, where $V$ is a vector space over a field $F$, and $\overline{F}$ denotes an algebraic closure of $F$. Let's not assume $G$ is finite, or that $V$ is finite dimensional, or anything about the characteristic of $F$.

Suppose that every $G$-linear map $V \rightarrow V$ is given by scalar multiplication. Does it hold that $\overline{\rho}: G \rightarrow \textrm{GL}(V \otimes_F \overline{F})$ remains irreducible?

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You can prove this using Jacobson's Density Theorem, which states that if $V$ is an irreducible module for a ring $R$, and $v_1,\dots,v_n$ are finitely many elements of $V$ that are linearly independent over $\text{End}_R(V)$, and $u_1,\dots,u_n\in V$, then there is some $r\in R$ with $v_1r=u_1,\dots,v_nr=u_n$.

In the situation of the question, with $R=FG$, the condition on endomorphisms of $V$ means that $v_1,\dots,v_n$ just need to be linearly independent over $F$.

Suppose $V\otimes_F\bar{F}$ were not irreducible. Choose any nonzero element $v=\sum_{i=1}^nv_i\otimes f_i$, where $v_1\dots,v_n$ are linearly independent over $F$, and all the $f_i\in\bar{F}$ are nonzero.

By Jacobson's Density Theorem, there is some $r \in FG$ with $v_1r=v_1, v_2r=\dots=v_nr=0$. So $vr=v_1$, which generates $V\otimes_F\bar{F}$.

Therefore $v$ generates $V\otimes_F\bar{F}$, and so, since $v$ is arbitrary, $V\otimes_F\bar{F}$ is irreducible.

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    Two questions: (1) are you using $v_1$ with two different meanings? You defined it in terms of an expansion of $v$ as $\sum v_i\otimes f_i$, but then you asserted it generates $V\otimes_F\overline{F}$, not sure why. (2) Doesn't the density theorem require that the $v_i$ can be simultaneously carried to the $u_i$ by an $\operatorname{End}_R(V)$-linear map? We don't have that here because the only $\operatorname{End}_R(V)$-linear maps are scalar multiplication. What am I missing? – 2017-02-25
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    (My understanding of the density theorem comes from its Wikipedia entry: https://en.wikipedia.org/wiki/Jacobson_density_theorem) – 2017-02-25
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    @BenBlum-Smith (1) $v_1$ generates $V$ as an $FG$-module (since $V$ is irreducible), and so generates $V\otimes_F\bar{F}$ as an $\bar{F}G$-module. – 2017-02-25
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    @BenBlum-Smith (2) The only $FG$-linear maps are scalar multiplication, so $\text{End}_{FG}(V)=F$ and *all* $F$-linear endomorphisms are $\text{End}_{FG}(V)$-linear. – 2017-02-25