A universal Clebsch-Gordan filtration for $\operatorname{GL}_{2,A}$
Helge Öystein Maakestad
TL;DR
The paper develops a universal Clebsch-Gordan filtration framework for $\operatorname{SL}_{2,A}$ and $\operatorname{GL}_{2,A}$ when $A$ is a Dedekind domain, extending classical results from fields of characteristic zero. It proves the existence of canonical filtrations with graded pieces given by symmetric powers, and derives a virtual Clebsch-Gordan formula in the Grothendieck group, clarifying how decomposition theorems fail over rings like $\mathbb{Z}$ while preserving a filtration structure. It also demonstrates the lack of complete reducibility for $\operatorname{SL}_{2,A}$ in several integral settings and develops a functorial theory of duals for comodules, including a transpose dual, with explicit constructions and examples. The work provides explicit nontrivial comodules with good filtrations, analyzes the second symmetric tensor and its dual, and offers a universal filtration for $\operatorname{GL}_{2,A}$ that specializes to the $\operatorname{SL}_{2,A}$ case, offering tools potentially applicable to flag bundles and vector bundles on quotient spaces. Overall, the results supply structural insight into representation theory over rings beyond fields, enabling refined algebraic and geometric applications.
Abstract
The aim of the paper is to study the group schemes $G:=\operatorname{SL}_{2, A}, \operatorname{GL}_{2,A}$ 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 $\operatorname{Sym}^n(V)$ for $n \geq 1$ an integer. If $A$ is a field of characteristic zero, there is a direct sum decomposition of the tensor product $\operatorname{Sym}^n(V) \otimes \operatorname{Sym}^m(V)$ 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 $1 \leq n \leq m$ a finite filtration $F_i \subseteq \operatorname{Sym}^n(V) \otimes \operatorname{Sym}^m(V)$ with $F_i/F_{i+1} \cong \operatorname{Sym}^{n+m-2i}(V)$ for $i=0,..,n$. 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 $\operatorname{GL}_{2,A}$. I moreover prove that the group scheme $G$ is not "completely reducible" in the sense that there are surjections $φ: V \rightarrow W$ 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 $(V, Δ)$, where $V$ is a free and finite rank $A$-module. This construction has the property that the double dual $V^{**}$ is canonically isomorphic to $V$ as comodule. I calculate some explicit examples.