Lemma. The restricted actions are the Clifford action [meinrenken2013clifford, Section 3.2.1, pp. 55--56] [fcap-001N]
Lemma. The restricted actions are the Clifford action [meinrenken2013clifford, Section 3.2.1, pp. 55--56] [fcap-001N]
For every \(g\in \operatorname {Spin}(Q)\) and \(h\in \operatorname {Pin}(Q)\), the two representations act through their underlying Clifford units: \[\operatorname {spinRep}(g)=\rho (g),\qquad \operatorname {pinRep}(h)=\rho (h).\] Thus the group actions introduce no new formula on \(S\); they are the same Clifford action with a smaller domain. In particular, any equation proved for the action of the underlying Clifford element applies to its Spin or Pin restriction.