Some remarks on Spin-orbits of unit vectors

TL;DR

This paper investigates the actions of special linear and spin groups on unimodular and unit vectors over certain rings, providing an example where the natural comparison map between their orbit spaces is not bijective.

Contribution

It presents a specific example of a ring where the comparison map between orbit spaces under SL_n(R) and Spin_{2n}(R) actions fails to be bijective.

Findings

The group actions are well-defined on unimodular and unit vectors.

A counterexample ring demonstrates the failure of the comparison map to be bijective.

The result highlights limitations in the relationship between these group actions.

Abstract

For $n \in \mathbb{N}$ and a commutative ring $R$ with $2 \in R^{\times}$, the group $SL_n (R)$ acts on the set $Um_n (R)$ of unimodular vectors of length $n$ and $Spin_{2n}(R)$ acts on the set of unit vectors $U_{2n-1}(R)$. We give an example of a ring for which the comparison map $Um_n (R)/SL_n (R) \rightarrow U_{2n-1}(R)/Spin_{2n}(R)$ fails to be bijective.