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

TL;DR

This paper investigates the structure of symmetric powers and Clebsch-Gordan filtrations for group schemes like SL_2 and GL_2 over rings, extending classical results from fields of characteristic zero to more general rings.

Contribution

It introduces a universal Clebsch-Gordan filtration for symmetric powers over rings, and explores the non-complete reducibility of group schemes, providing explicit constructions and functorial duality definitions.

Findings

Existence of a finite filtration with symmetric power quotients over rings

A version of the Clebsch-Gordan formula in the Grothendieck group

Group schemes are not completely reducible over rings

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 $\phi: V \rightarrow W$ 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 $(V, \Delta)$, 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.