Editing Spinor (section)

=== Exterior algebra construction ===
The computations with the minimal ideal construction suggest that a spinor representation can
also be defined directly using the [[exterior algebra]] {{math|1=Λ<sup>∗</sup> ''W'' = ⊕<sub>''j''</sub> Λ<sup>''j''</sup> ''W''}} of the isotropic subspace ''W''.
Let {{math|1=Δ = Λ<sup>∗</sup> ''W''}} 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.<ref>One source for this subsection is {{Harvtxt|Fulton|Harris|1991}}.</ref><ref>Jurgen Jost, "Riemannian Geometry and Geometric Analysis" (2002) Springer-Verlag Universitext {{ISBN|3-540-42627-2}}. ''See chapter 1.''</ref>

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 [[Clifford algebra#Universal property and construction|universal property of Clifford algebras]]. The details differ slightly according to whether the dimension of ''V'' is even or odd.

When dim({{mvar|V}}) is even, {{math|1=''V'' = ''W'' ⊕ ''W''{{′}}}} where ''W''{{′}} is the chosen isotropic complement. Hence any {{math|''v'' ∈ ''V''}} decomposes uniquely as {{math|1=''v'' = ''w'' + ''w''{{′}}}} with {{math|''w'' ∈ ''W''}} and {{math|''w''{{′}} ∈ ''W''{{′}}}}. The action of {{mvar|v}} on a spinor is given by
<math display="block">c(v) w_1 \wedge\cdots\wedge w_n = \left(\epsilon(w) + i\left(w'\right)\right)\left(w_1 \wedge\cdots\wedge w_n\right)</math>
where ''i''(''w''{{′}}) is [[interior product]] with ''w''{{′}} using the nondegenerate quadratic form to identify ''V'' with ''V''<sup>∗</sup>, and ''ε''(''w'') denotes the [[exterior product]]. This action is sometimes called the '''Clifford product'''. It may be verified that
<math display="block">c(u)\,c(v) + c(v)\,c(u) = 2\,g(u,v)\,,</math>
and so {{mvar|c}} 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<ref>Via the even-graded Clifford algebra.</ref> (the half-spin representations, or Weyl spinors) via
<math display="block">\Delta_+ = \Lambda^\text{even} W,\, \Delta_- = \Lambda^\text{odd} W.</math>

When dim(''V'') is odd, {{math|1=''V'' = ''W'' ⊕ ''U'' ⊕ ''W''{{′}}}}, where ''U'' is spanned by a unit vector ''u'' orthogonal to ''W''. The Clifford action ''c'' is defined as before on {{math|''W'' ⊕ ''W''′}}, while the Clifford action of (multiples of) ''u'' is defined by
<math display="block">c(u)\alpha = \begin{cases}
   \alpha & \hbox{if } \alpha \in \Lambda^\text{even} W \\
  -\alpha & \hbox{if } \alpha \in \Lambda^\text{odd} W
\end{cases}</math>
As before, one verifies that ''c'' respects the Clifford relations, and so induces a homomorphism.