Editing Spinor (section)

=== Even dimensions ===
If {{math|1=''n'' = 2''k''}} is even, then the tensor product of Δ with the [[contragredient representation]] decomposes as
<math display="block">\Delta\otimes\Delta^* \cong \bigoplus_{p=0}^n \Gamma_p \cong \bigoplus_{p=0}^{k-1} \left(\Gamma_p\oplus\sigma\Gamma_p\right) \oplus \Gamma_k</math>
which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elements {{math|''αω'' ⊗ ''βω''′}}. The rightmost formulation follows from the transformation properties of the [[Hodge star operator]]. Note that on restriction to the even Clifford algebra, the paired summands {{math|Γ<sub>''p''</sub> ⊕ ''σ''Γ<sub>''p''</sub>}} are isomorphic, but under the full Clifford algebra they are not.

There is a natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra:
<math display="block" display="block">(\alpha\omega)^* = \omega\left(\alpha^*\right).</math>
So {{math|Δ ⊗ Δ}} also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose
<math display="block">\begin{align}
  \Delta_+\otimes\Delta^*_+ \cong \Delta_-\otimes\Delta^*_- &\cong \bigoplus_{p=0}^k \Gamma_{2p}\\
  \Delta_+\otimes\Delta^*_- \cong \Delta_-\otimes\Delta^*_+ &\cong \bigoplus_{p=0}^{k-1} \Gamma_{2p+1}
\end{align}</math>

For the complex representations of the real Clifford algebras, the associated [[reality structure]] on the complex Clifford algebra descends to the space of spinors (via the explicit construction in terms of minimal ideals, for instance). In this way, we obtain the complex conjugate {{overline|Δ}} of the representation Δ, and the following isomorphism is seen to hold:
<math display="block">\bar{\Delta} \cong \sigma_-\Delta^*</math>

In particular, note that the representation Δ of the orthochronous spin group is a [[unitary representation]]. In general, there are Clebsch–Gordan decompositions
<math display="block">\Delta \otimes\bar{\Delta} \cong \bigoplus_{p=0}^k\left(\sigma_-\Gamma_p \oplus \sigma_+\Gamma_p\right).</math>

In metric signature {{math|(''p'', ''q'')}}, the following isomorphisms hold for the conjugate half-spin representations
* If ''q'' is even, then <math>\bar{\Delta}_+ \cong \sigma_- \otimes \Delta_+^*</math> and <math>\bar{\Delta}_- \cong \sigma_- \otimes \Delta_-^*.</math>
* If ''q'' is odd, then <math>\bar{\Delta}_+ \cong \sigma_- \otimes \Delta_-^*</math> and <math>\bar{\Delta}_- \cong \sigma_- \otimes \Delta_+^*.</math>
Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the half-spin representations {{math|Δ<sub>±</sub> ⊗ {{overline|Δ}}<sub>±</sub>}}.