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In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group.

In general the map from the Pin group to the orthogonal group is not onto or a universal covering space, but if the quadratic form is definite, it is both.

The pin group of a definite form maps onto the orthogonal group, and each component is simply connected: it double covers the orthogonal group. The pin groups for a positive definite quadratic form $Q$ and for its negative $-Q$ are not isomorphic, but the orthogonal groups are.

In terms of the standard forms, $O(n,0)\; =\; O(0,n)$, but $mbox\{Pin\}(n,0)\; notcong\; mbox\{Pin\}(0,n)$. Using the "+" sign convention for Clifford algebras (where $v^2=Q(v)\; in\; Cell(V,Q)$), one writes

- $mbox\{Pin\}\_+(n)\; :=\; mbox\{Pin\}(n,0)\; qquad\; mbox\{Pin\}\_-(n)\; :=\; mbox\{Pin\}(0,n)$

By contrast, we have the isomorphism $mbox\{Spin\}(n,0)\; cong\; mbox\{Spin\}(0,n)$ and they are both the (unique) universal cover of the special orthogonal group SO(n).

The Pin and Spin groups are particular topological groups associated to the orthogonal and special orthogonal groups, coming from Clifford algebras: there are other similar groups, corresponding to other double covers or to other group structures on the other components, but they are not referred to as Pin or Spin groups, nor studied much.

- $1\; to\; \{pm\; 1\}\; to\; mbox\{Pin\}\_pm(V)\; to\; O(V)\; to\; 1$

The two extensions are distinguished by whether the preimage of a reflection squares to $pm\; 1\; in\; ker\; left(mbox\{Spin\}(V)\; to\; SO(V)right)$, and the two pin groups are named accordingly. Explicitly, a reflection has order 2 in $O(V)$, $r^2=1$, so the square of the preimage of a reflection (which has determinant one) must be in the kernel of $mbox\{Spin\}\_pm(V)\; to\; SO(V)$, so $tilde\; r^2\; =\; pm\; 1$, and either choice determines a pin group (since all reflections are conjugate by an element of $SO(V)$, which is connected, all reflections must square to the same value).

Concretely, in $mbox\{Pin\}\_+$, $tilde\; r$ has order 2, and the preimage of a subgroup $\{1,r\}$ is $C\_2\; times\; C\_2$: if one repeats the same reflection twice, one gets the identity.

In $mbox\{Pin\}\_-$, $tilde\; r$ has order 4, and the preimage of a subgroup $\{1,r\}$ is $C\_4$: if one repeats the same reflection twice, one gets "a rotation by 2π"—the non-trivial element of $mbox\{Spin\}(V)\; to\; SO(V)$ can be interpreted as "rotation by 2π" (every axis yields the same element).

In $mbox\{Pin\}\_+$, the preimage of the dihedral group of an $n$-gon, considered as a subgroup $mbox\{Dih\}\_n\; <\; O(2)$, is the dihedral group of an $2n$-gon, $mbox\{Dih\}\_\{2n\}\; <\; mbox\{Pin\}\_+(2)$, while in $mbox\{Pin\}\_-$, the preimage of the dihedral group is the dicyclic group $mbox\{Dic\}\_n\; <\; mbox\{Pin\}\_-(2)$.

In 1 dimension, the pin groups are congruent to the first dihedral and dicyclic groups:

- $begin\{align\}$

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Last updated on Tuesday November 27, 2007 at 16:15:21 PST (GMT -0800)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday November 27, 2007 at 16:15:21 PST (GMT -0800)

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