Editing Spinor (section)

=== Spinors in representation theory ===
{{Main|Spin representation}}

One major mathematical application of the construction of spinors is to make possible the explicit construction of [[linear representation]]s of the [[Lie algebra]]s of the [[special orthogonal group]]s, and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the [[Atiyah–Singer index theorem]], and to provide constructions in particular for [[discrete series]] representations of [[semisimple group]]s.

The spin representations of the special orthogonal Lie algebras are distinguished from the [[tensor]] representations given by [[Young symmetrizer|Weyl's construction]] by the [[weight (representation theory)|weights]]. Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the [[spin representation]] article.