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definition. Spin group [garling2011clifford] [spin-0008]
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     Suppose that \((E, q)\) is a regular quadratic space. We consider the action of \(\mathcal {G}(E, q)\) on \(\mathcal {A}(E, q)\) by adjoint conjugation. We set \[ A d_g^{\prime }(a)=g a g^{-1}, \] for \(g \in \mathcal {G}(E, q)\) and \(a \in \mathcal {A}(E, q)\). We restrict attention to those elements of \(\mathcal {G}(E, q)\) which stabilize \(E\). The Clifford group \(\Gamma =\Gamma (E, q)\) is defined as \[ \left \{g \in \mathcal {G}(E, q): A d_g^{\prime }(x) \in E \text { for } x \in E\right \} . \] If \(g \in \Gamma (E, q)\), we set \(\alpha (g)(x)=A d_g^{\prime }(x)\). Then \(\alpha (g) \in G L(E)\), and \(\alpha \) is a homomorphism of \(\Gamma \) into \(G L(E) . \alpha \) is called the graded vector representation of \(\Gamma \). It is customary to scale the elements of \(\Gamma (E, q)\); we set \[ \begin {aligned} \operatorname {Pin}(E, q) & =\{g \in \Gamma (E, q): \Delta (g)= \pm 1\}, \\ \Gamma _1(E, q) & =\{g \in \Gamma (E, q): \Delta (g)=1\} . \end {aligned} \] If \((E, q)\) is a Euclidean space, then \(\operatorname {Pin}(E, q)=\Gamma _1(E, q)\); otherwise, \(\Gamma _1(E, q)\) is a subgroup of \(\operatorname {Pin}(E, q)\) of index 2. We have a short exact sequence \[ 1 \longrightarrow D_2 \xrightarrow {\subseteq } \operatorname {Pin}(E, q) \xrightarrow {\alpha } O(E, q) \longrightarrow 1 ; \] \(\operatorname {Pin}(E, q)\) is a double cover of \(O(E, q)\). In fact there is more interest in the subgroup \(\operatorname {Spin}(E, q)\) of \(\operatorname {Pin}(E, q)\) consisting of products of an even number of unit vectors in \(E\). Thus \(\operatorname {Spin}(E, q)=\operatorname {Pin}(E, q) \cap \mathcal {A}^{+}(E, q)\) and \[ \operatorname {Spin}(E, q)=\left \{g \in \mathcal {A}^{+}(E, q): g E=E g \text { and } \Delta (g)= \pm 1\right \} . \] If \(x, y\) are unit vectors in \(E\) then \(\alpha (x y)=\alpha (x) \alpha (y) \in S O(E, q)\), so that \(\alpha (\operatorname {Spin}(E, q)) \subseteq S O(E, q)\). Conversely, every element of \(S O(E, q)\) is the product of an even number of simple reflections, and so \(S O(E, q) \subseteq \alpha \left (\operatorname {Spin}(E, q)\right )\). Thus \(\alpha \left (\operatorname {Spin}(E, q)\right )=S O(E, q)\), and we have a short exact sequence. \[ 1 \longrightarrow D_2 \xrightarrow {\subseteq } \operatorname {Spin}(E, q) \xrightarrow {\alpha } S O(E, q) \longrightarrow 1 ; \] \(\operatorname {Spin}(E, q)\) is a double cover of \(S O(E, q)\). Note also that if \(a \in \operatorname {Spin}(E, q)\) and \(x \in E\) then \(\alpha (a)(x)=a x a^{-1}\); conjugation and adjoint conjugation by elements of \(\operatorname {Spin}(E, q)\) are the same.