has characteristic . Similarly there is a homomorphism from the Spin group to the special orthogonal group of . In the common case when is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when has dimension at least . Further the kernel of this homomorphism consists of and . So in this case the spin group, , is a double cover of . Please note, however, that the simple connectedness of the spin group is not true in general: if is for and both at least then the spin group is not simply connected. In this case the algebraic group is simply connected as an algebraic group, even though its group of real valued points is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups. Spinors Clifford algebras , with even, are matrix algebras which have a complex representation of dimension . By restricting to the group 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 then it splits as the sum of two half spin representations (or Weyl representations) of dimension . If is odd then the Clifford algebra is a sum of two matrix algebras, each of which has a representation of dimension , and these are also both representations of the Pin group . On restriction to the spin group these become isomorphic, so the spin group has a complex spinor representation of dimension . 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. Real spinors To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The pin group, is the set of invertible elements in that can be written as a product of unit vectors: Comparing with the above concrete realizations of the Clifford algebras, the pin group corresponds to the products of arbitrarily many reflections: it