If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over $\mathrm{Spec}(\mathbf{Z})$? Or is life not so easy? I feel like if I actually knew the construction of the irreps of $G_{\mathbf{C}}$ (rather than treating the entire "highest weight" theory as a black box, which I've done up to this point in my life) I should be able to answer this.

For $\mathrm{SL}(2)$ the answer is going to be yes, because the only irreps are symmetric powers, which can be realised over $\mathbf{Z}$. But for groups of rank greater than 1 I realise I am a bit unclear about the relationship between irreps (of the algebraic group) in char 0 and in char $p$; for example can the dimension of the irrep with some given highest weight jump as one moves from char 0 to char $p$? Even if it can, this doesn't mean my question has a negative answer.