A universal Clebsch-Gordan filtration for ${\mathrm{GL}}_{2,A}$

The aim of the paper is to study the group schemes $G:={\mathrm{SL}}_{2,A},{\mathrm{GL}}_{2,A}$ and universal Clebsch-Gordan filtrations. Here $A$ is a field or any commutative ring. If $V:=A\{e_1,e_2\}$ is the free rank $2$ module on $A$ and if we give $V$ the “standard” structure as comodule on $G$, we may form the symmetric powers ${\mathrm{Sym}}^n(V)$ for $n\ge 1$ an integer. If $A$ is a field of characteristic zero, there is a direct sum decomposition of the tensor product ${\mathrm{Sym}}^n(V)\otimes {\mathrm{Sym}}^m(V)$ into irreducible $G$-comodules and the main aim of the paper is to investigate if similar results hold over the ring of integers or a more general commutative ring such as a Dedekind domain. For $A:=\mathbb{Z}$ we will find that there is for any pair of integers $1\le n\le m$ a finite filtration $F_i\subseteq {\mathrm{Sym}}^n(V)\otimes {\mathrm{Sym}}^m(V)$ with $F_i\mathrm{/}{F}_{i+1}\cong {\mathrm{Sym}}^{n+m-2i}(V)$ for $i=0,\mathrm{.}\mathrm{.},n$. This implies there is a version of the Clebsch-Gordan formula valid in the Grothendieck group of coherent comodules on $G$. I also prove a similar result for ${\mathrm{GL}}_{2,A}$. I moreover prove that the group scheme $G$ is not “completely reducible” in the sense that there are surjections $\phi :V\to W$ of finite rank comodules on $G$ that do not split. I also discuss the notion “good filtration” for torsion free comodules and give an explicit construction of an infinte set of non trival comodules with a good filtration. I give a functorial definition of the dual comodule of any comodule $(V,\mathrm{\Delta })$, where $V$ is a free and finite rank $A$-module. This construction has the property that the double dual $V^{\ast \ast }$ is canonically isomorphic to $V$ as comodule. I calculate some explicit examples.