Some remarks on Spin-orbits of unit vectors

Tariq Syed

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arXiv · 2308.11120

Some remarks on Spin-orbits of unit vectors

Tariq Syed

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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.

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Tariq Syed. 2023-08-22. Some remarks on Spin-orbits of unit vectors. https://arxiv.org/abs/2308.11120

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