Spinors
Clifford algebras Cℓp,q(C), with p+q=2n even, are matrix algebras which have a complex representation of dimension 2n. By restricting to the group Pinp,q(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 Spinp,q(R) then it splits as the sum of two half spin representations (or Weyl representations) of dimension 2n−1.
If p+q=2n+1 is odd then the Clifford algebra Cℓp,q(C) is a sum of two matrix algebras, each of which has a representation of dimension 2n, and these are also both representations of the Pin group Pinp,q(R). On restriction to the spin group Spinp,q(R) these become isomorphic, so the spin group has a complex spinor representation of dimension 2n.
More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the 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 spinors.
Read more about this topic: Clifford Algebra