Let $G$ be an affine group scheme. Let $\pi : P\rightarrow X$ be a principal $G$-bundle over a scheme $X$ ( i.e. $\pi$ is surjective flat affine morphism, $\phi : P\times G\rightarrow P$ action of $G$ on $P$ such that $\pi \circ \phi = \pi \circ p_1$; where $p_1 : P\times G\rightarrow P$, and $\psi : P\times G\rightarrow P\times_X P$ defined by $p_1\times \phi$ is an isomorphism).
Let $V$ be a finite dimensional representation. Then How to get vector bundle (locally free sheaf of finite rank) on $X$ associated to $P$ and $V$. In the setting of manifold I know this, but in scheme setting, we can not take point wise action. I don't know any good reference for this.