complex vs modular representation theory Let $G$ be a finite group and consider the representation theory of $G$ over the algebraic closure $\overline{\mathbf{F}_p}$ of a finite field with $p$ elements, where $p$ is coprime to $|G|$. It is classical that in this situation, the representation theory of $G$ is "for all intents and purposes" equivalent to that over the complex field $\mathbf{C}$. I am wondering if there is a precise form of this statement. To be concrete, is there an equivalence between the category of finite dimensional representations of $G$ over $\overline{\mathbf{F}_p}$ and over $\mathbf{C}$ that respects irreducibility and direct sums? How are the characters related?
 A: No, there cannot be such an equivalence, because the category of finite-dimensional representations of $G$ over a field $K$ is sensitive to $K$: the endomorphism ring of every irreducible representation is a finite-dimensional division algebra over $K$, so in particular this category knows the characteristic of $K$, and if $K$ is algebraically closed then this category knows $K$ itself.
What I believe is true but have not carefully checked is, as Jyrki suggests in the comments, that there is some ring of integers $\mathcal{O}_K$ in some number field $K$ (which can be chosen to be a cyclotomic field), admitting homomorphisms to both $\mathbb{C}$ and $\overline{\mathbb{F}_p}$ such that the corresponding extension of scalars functors $\text{Rep}_{\mathcal{O}_K}(G) \to \text{Rep}_{\mathbb{C}, \overline{\mathbb{F}_p}}(G)$

*

*are symmetric monoidal; this implies that they respect traces and characters in the sense that the trace / character over $\mathbb{C}, \overline{\mathbb{F}_p}$ is the image of the trace / character over $\mathcal{O}_K$ with respect to the above homomorphisms,

*induce bijections on isomorphism classes of representations, and

*are additive and have the property that their hom spaces are the extensions of scalars of the hom spaces over $\mathcal{O}_K$.

Edit: This is not quite right; at a minimum we need to take $\text{Rep}_{\mathcal{O}_K}(G)$ to consist only of representations of $G$ on finite free $\mathcal{O}_K$-modules, and even then I'm not actually sure we get a bijection on isomorphism classes. Also we may need to take the localization $\mathcal{O}_K[|G|^{-1}]$ to be safe.
This is as close to an equivalence as I think you can get. I don't know a really slick proof of this; you need to show that every representation of $G$ is defined over the algebraic integers (actually it would be fine if we inverted $|G|$) which IIRC is a bit of a slog. The easiest special case of this occurs with the symmetric groups where it's classically known that all of their irreducible representations are defined over $\mathbb{Z}$ so you can take $\mathcal{O}_K = \mathbb{Z}$ here.
Edit #2: I've asked a question about this.
Edit #3: In the linked question you can find an argument which shows that there exists a finitely presented commutative ring $R_{\text{Iso}}$ with the property that

*

*Every irreducible representation of $G$ over $\mathbb{C}$ is defined over $R_{\text{Iso}}$, and

*$R_{\text{Iso}}$ admits homomorphisms to $\mathbb{C}$ and to $\overline{\mathbb{F}_p}$ for all primes $p \nmid |G|$ such that extension of scalars of the representations constructed above along these homomorphisms reproduces the irreducible representations of $G$ over these fields. In particular, the character of these representations can be computed over $R_{\text{Iso}}$ and maps to the character over $\mathbb{C}$ and over $\overline{\mathbb{F}_p}$.

This is pretty close to satisfying. I think it is furthermore true that the category of representations of $G$ on finite free $R_{\text{Iso}}$-modules is close to equivalent to the category of finite-dimensional representations of $G$ over $\mathbb{C}$ and over $\overline{\mathbb{F}_p}$ in the above sense but I haven't checked this carefully.
