I am trying to solve the following problem (3.3 in Waterhouse's Introduction to Affine Group Schemes):
Let $\underline{\Gamma}$ be a finite constant group scheme over a field $k$ (corresponding to an abstract finite group $\Gamma$). Show that $n$-dimensional linear representations of $\underline{\Gamma}$ are given by ordinary [group] homomorphisms $\underline\Gamma(k)\to GL_n (k)$.
I have considered two ways to think about this problem, although I want to prove it in a functorial manner (the way the book has done so far). But first, a geometric way that I think can be made rigorous. As a scheme, $\underline\Gamma$ is just a finite disjoint union of copies of $\text{Spec}(k)$, with the group operation induced from the original underlying group $\Gamma$, so the only points of $\underline\Gamma$ are $k$-points. Any homomorphism of $k$-schemes $\phi: \underline\Gamma\to GL_n$ (i.e. a representation of $\underline\Gamma$) takes $k$-points to $k$-points, and because $\underline\Gamma$ consists entirely of $k$-points, $\phi$ is completely determined by the induced map $\underline\Gamma(k)\cong\Gamma\to GL_n (k)$. So geometrically, a representation of a finite constant group scheme over $k$ should be determined by its induced representation on rational points only.
Thinking in a functorial way though, I have a more sketchy answer that I want to clarify. We have a bijection on objects that I hope we can refine to an equivalence:
$$(\text{Reps of } \underline\Gamma)\simeq(k^\Gamma \text{-comodules}),$$
where $k^\Gamma = \prod_{g\in\Gamma} k$ has a Hopf algebra structure and represents the functor $\underline\Gamma$. Now the dual algebra of $k^\Gamma$ is $k[\Gamma]$, and I would hope there's an equivalence
$$(k^\Gamma\text{-comodules})\simeq(k[\Gamma]\text{-modules}),$$
although I'm not sure this is right. The latter category is equivalent to the category of $k$-representations of the group $\Gamma$, and we have $\Gamma\cong\underline\Gamma(k)$ since $\text{Spec}(k)$ is connected. Therefore, we have
$$(\text{Reps of the group scheme }\underline\Gamma)\simeq (\text{Reps of the group } \underline{\Gamma}(k)).$$
I quite like this result, due to the mixed geometric/categorical flavour, as well as this sketch "proof", but is it correct? Have I made any faulty assumptions (for example, the equivalence of categories of (co)modules)? And is there another way to prove/think about this result? Many thanks for your help!
Your equivalence of categories is correct.
If $A$ is a finite dimensional vector space over a field $k$, and $A^\ast$ is the dual, then $-\otimes A$ is right adjoint to $-\otimes A^\ast$, so for any vector space $M$, maps $\alpha:M\to M\otimes A$ correspond to maps $\beta:M\otimes A^\ast\to M$. If $A$ is a Hopf algebra (or even just a coalgebra) then it's straightforward to check that $\alpha$ makes $M$ into an $A$-comodule if and only if $\beta$ makes $M$ into an $A^\ast$-module.