Let $S$ be a scheme and $\mathcal A$ a quasicoherent $\mathcal O_S$-Algebra.
One knows that then one can associate the affine $S-$scheme $Spec(\mathcal A)$ over $S$.
In particular I can consider $Spec(Sym(\mathcal E))$ for a quasicoherent sheaf $\mathcal E$ on $S$, where $Sym$ denotes the symmetric algebra of the sheaf.
My question is:
Is there a natural structure of a $S-$group scheme on $Spec(Sym(\mathcal E))$? At least if $\mathcal E$ is locally free of finite rank, this should be true.
One could argue that one just glues the local addition maps as locally on $S$ the bundle is just affine $n-$space.
But I would be interested in what group functor it represents.