Spinor norm

Definition

Suppose $k$ is a field, $V$ is a vector space over $k$, and $b$ is a nondegenerate symmetric bilinear form. Let $G$ be the orthogonal group corresponding to $b$. The spinor norm is a homomorphism from $G$ to the multiplicative group modulo squares of $k$:

$\!G\to k^{*}/(k^{*})^{2}$

i.e., it is a homomorphism that sends any element in the orthogonal group to an element in the multiplicative group of $k$ modulo the squares in that group.

The homomorphism is defined as follows: any element of $G$ arising as reflection orthogonal to a vector $v$ is sent to the value $b(v,v)$ modulo $(k^{*})^{2}$. This extends to a well defined and unique homomorphism on all of $G$. For characteristic not equal to 2, the reflection orthogonal to $v$ is defined as:

$\!x\mapsto x-2v{\frac {b(v,x)}{b(v,v)}}$

For characteristic equal to 2, the reflection orthogonal to $v$ is defined as:

$\!x\mapsto x-v{\frac {b(v,x)}{b(v,v)}}$

Also note that for such a reflection map to exist, $b(v,v)$ must be nonzero, so the map does indeed go to $(k^{*})/(k^{*})^{2}$.

Note that different choices of $v$ that are scalar multiples of each other define the same reflection map. That is why the spinor norm is defined only as a map to $k^{*}/(k^{*})^{2}$ and not as a map to $k^{*}$.

Particular cases

• For a Pythagorean field (i.e., a field where a sum of squares is a square), the spinor norm for the orthogonal group for the standard dot product is trivial. In particular, this applies to the orthogonal group for the standard dot product over the field of real numbers.
• For an algebraically closed field, the spinor norm for any symmetric bilinear form is trivial because $k^{*}/(k^{*})^{2}$ is trivial.