Pin group

Example

Let $n \ge 1$, let $\varepsilon \in \set{1, -1}$, and let $\mathrm{Cl}$ be the real Clifford algebra of $\R^n$ with the relations $v^2 = \varepsilon \abs{v}^2$ for $v \in \R^n$. Let $\alpha$ be the automorphism of $\mathrm{Cl}$ with $\alpha(v) = -v$ for $v \in \R^n$. The pin group is the set of all products $u_1 u_2 \cdots u_k$ in $\mathrm{Cl}$ of unit vectors $u_i \in \R^n$, with any number $k \ge 0$ of factors. It is a group under the Clifford product, since $u_i^{-1} = \varepsilon u_i$.

For $x$ in the pin group and $v \in \R^n$, set $\rho(x)(v) = \alpha(x) v x^{-1}$. Check for one unit vector $u$: $uv = -vu$ when $v \perp u$, so $\rho(u)(v) = -u v u^{-1}$ equals $v$ for $v \perp u$ and $-u$ for $v = u$. So $\rho(u)$ is the reflection of $\R^n$ in the hyperplane $u^\perp$. Since every element of the orthogonal group $O(n)$ is a product of reflections, $\rho$ is a surjective homomorphism onto $O(n)$, and its kernel is $\set{1, -1}$.