Editing Clifford algebra (section)

== Spinors ==
<!-- This section is linked from [[Spinor]] -->
Clifford algebras {{math|Cl{{sub|''p'',''q''}}('''C''')}}, with {{math|1=''p'' + ''q'' = 2''n''}} even, are matrix algebras that have a complex representation of dimension {{math|2<sup>''n''</sup>}}. By restricting to the group {{math|Pin<sub>''p'',''q''</sub>('''R''')}} we get a complex representation of the Pin group of the same dimension, called the [[spin representation]]. If we restrict this to the spin group {{math|Spin<sub>''p'',''q''</sub>('''R''')}} then it splits as the sum of two ''half spin representations'' (or ''Weyl representations'') of dimension&nbsp;{{math|2<sup>''n''−1</sup>}}.

If {{math|1=''p'' + ''q'' = 2''n'' + 1}} is odd then the Clifford algebra {{math|Cl{{sub|''p'',''q''}}('''C''')}} is a sum of two matrix algebras, each of which has a representation of dimension {{math|2<sup>''n''</sup>}}, and these are also both representations of the pin group {{math|Pin<sub>''p'',''q''</sub>('''R''')}}. On restriction to the spin group {{math|Spin<sub>''p'',''q''</sub>('''R''')}} these become isomorphic, so the spin group has a complex spinor representation of dimension&nbsp;{{math|2<sup>''n''</sup>}}.

More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the [[classification of Clifford algebras|structure of the corresponding Clifford algebras]]: whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra.
For examples over the reals see the article on [[spinor]]s.

=== Real spinors ===
{{details|spinor}}
To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The [[pin group]], {{math|Pin<sub>''p'',''q''</sub>}} is the set of invertible elements in {{math|Cl{{sub|''p'',''q''}}}} that can be written as a product of unit vectors:
<math display="block">\mathrm{Pin}_{p,q} = \left\{v_1v_2 \cdots v_r \mid \forall i\, \|v_i\| = \pm 1\right\}.</math>
Comparing with the above concrete realizations of the Clifford algebras, the pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group {{math|O(''p'', ''q'')}}.  The [[spin group]] consists of those elements of {{math|Pin<sub>''p'',''q''</sub>}} that are products of an even number of unit vectors.  Thus by the [[Cartan–Dieudonné theorem]] Spin is a cover of the group of proper rotations {{math|SO(''p'', ''q'')}}.

Let {{math|''α'' : Cl → Cl}} be the automorphism that is given by the mapping {{math|''v'' ↦ −''v''}} acting on pure vectors.  Then in particular, {{math|Spin<sub>''p'',''q''</sub>}} is the subgroup of {{math|Pin<sub>''p'',''q''</sub>}} whose elements are fixed by {{math|''α''}}.  Let
<math display="block">\operatorname{Cl}_{p,q}^{[0]} = \{ x\in \operatorname{Cl}_{p,q} \mid \alpha(x) = x\}.</math>
(These are precisely the elements of even degree in {{math|Cl{{sub|''p'',''q''}}}}.)  Then the spin group lies within {{math|Cl{{su|lh=0.9em|p=[0]|b=''p'',''q''}}}}.

The irreducible representations of {{math|Cl{{sub|''p'',''q''}}}} restrict to give representations of the pin group.  Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner.  Thus the two representations coincide.  For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of {{math|Cl{{su|lh=0.9em|p=[0]|b=''p'',''q''}}}}.

To classify the pin representations, one need only appeal to the [[classification of Clifford algebras]].  To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above)
<math display="block">\operatorname{Cl}^{[0]}_{p,q} \approx \operatorname{Cl}_{p,q-1}, \text{ for } q > 0</math>
<math display="block">\operatorname{Cl}^{[0]}_{p,q} \approx \operatorname{Cl}_{q,p-1}, \text{ for } p > 0</math>
and realize a spin representation in signature {{math|(''p'', ''q'')}} as a pin representation in either signature {{math|(''p'', ''q'' − 1)}} or {{math|(''q'', ''p'' − 1)}}.