definition. Spin group [jadczyk2019notes] [spin-0007]
definition. Spin group [jadczyk2019notes] [spin-0007]
We define the Clifford group \(\Gamma =\Gamma (q)\) to be the group of all invertible elements \(u \in \mathrm {Cl}(q)\) which have the property that \(uyu^{-1}\) is in \(M\) whenever \(y\) is in \(M\). We define \(\Gamma (q)^{ \pm }\)as the intersection of \(\Gamma (q)\) and \(\mathrm {Cl}(q)_{ \pm }\).
For every element \(u \in \Gamma (q)\) we define the spinor norm \(N(u)\) by the formula \[ N(u)=\tau (u) u, \] where \(\tau \) is the main involution of the Clifford algebra \(\mathrm {Cl}(q)\). The following groups are called spin groups: \[ \begin {aligned} & \operatorname {Pin}(q):=\left \{s \in \Gamma (q)^{+} \cup \Gamma (q)^{-}: N(s)= \pm 1\right \} \\ & \operatorname {Spin}(q):=\left \{s \in \Gamma (q)^{+}: N(s)= \pm 1\right \} \\ & \operatorname {Spin}^{+}(q):=\left \{s \in \Gamma (q)^{+}: N(s)=+1\right \} . \end {aligned} \]