Spinor norm

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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