expansion of α with maximal homogeneous degree in the elements wi: (no summation implied), then is a nonzero scalar multiple of ω, as required. Note that for n even, this computation also shows that as a vector space. In the last equality we again used that W is isotropic. In physics terms, this shows that Δ is built up like a Fock space by creating spinors using anti-commuting creation operators in W acting on a vacuum ω. Exterior algebra construction The computations with the minimal ideal construction suggest that a spinor representation can also be defined directly using the exterior algebra of the isotropic subspace W. Let denote the exterior algebra of W considered as vector space only. This will be the spin representation, and its elements will be referred to as spinors. The action of the Clifford algebra on Δ is defined first by giving the action of an element of V on Δ, and then showing that this action respects the Clifford relation and so extends to a homomorphism of the full Clifford algebra into the endomorphism ring End(Δ) by the universal property of Clifford algebras. The details differ slightly according to whether the dimension of V is even or odd. When dim() is even, where W is the chosen isotropic complement. Hence any decomposes uniquely as with and . The action of on a spinor is given by where i(w) is interior product with w using the nondegenerate quadratic form to identify V with V∗, and ε(w) denotes the exterior product. This action is sometimes called the Clifford product. It may be verified that and so respects the Clifford relations and extends to a homomorphism from the Clifford algebra to End(Δ). The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group (the half-spin representations, or Weyl spinors) via When dim(V) is odd, , where U is spanned by a unit vector u orthogonal to W. The Clifford action c is defined as before on , while the Clifford action of (multiples of) u is defined by As before, one verifies that c respects the Clifford relations, and so induces a homomorphism. Hermitian vector spaces and spinors If the vector space V has extra structure that provides a decomposition of its complexification into two maximal isotropic subspaces, then the definition of spinors (by either method) becomes natural. The main example is