definition. Spin group [hahn2004clifford] [spin-000J]
definition. Spin group [hahn2004clifford] [spin-000J]
We continue to let \(F\) be a field of characteristic not 2 and \(M\) a quadratic space over \(F\).
Recall that \(\gamma : M \rightarrow C(M)\) is injective and that there is a unique involution - on \(C(M)\) taking \(\gamma x\) to \(\gamma x\) for all \(x\). Consider \(M\) to be a subset of \(C(M)\) via \(\gamma \), and define the group \(\operatorname {Spin}(M)\) by \[ \operatorname {Spin}(M)=\left \{c \in C_0(M)^{\times } \mid c M c^{-1}=M, c \bar {c}=1_C\right \}, \] where \(C_0(M)^{\times }\)is the group of invertible elements of the \(\operatorname {ring} C_0(M)\). The isometries from \(M\) onto \(M\) constitute the orthogonal group \(O(M)\) and \(S O(M)\) is the subgroup of elements of determinant 1. For \(c\) in \(\operatorname {Spin}(M)\), define \[ \pi c: M \rightarrow M \] by \(\pi c(x)=c x c^{-1}\). This provides a homomorphism \[ \pi : \operatorname {Spin}(M) \rightarrow S O(M) . \] By a theorem of Cartan and Dieudonné, any element \(\sigma \) in \(O(M)\) is a product \(\sigma =\tau _{y_1} \cdots \tau _{y_k}\) of hyperplane reflections \(\tau _{y_i}\). The assignment \(\Theta (\sigma )=\) \(q\left (y_1\right ) \cdots q\left (y_k\right )\left (F^{\times }\right )^2\) defines the spinor norm homomorphism \[ \Theta : S O(M) \rightarrow F^{\times } /\left (F^{\times }\right )^2 . \]