If n = 2 k is even, then the tensor product of Δ with the contragredient representation decomposes as Δ Δ ⊗ ⊗ Δ Δ ∗ ∗ ≅ ≅ ⨁ ⨁ p = 0 n Γ Γ p ≅ ≅ ⨁ ⨁ p = 0 k − − 1 ( Γ Γ p ⊕ ⊕ σ σ Γ Γ p ) ⊕ ⊕ Γ Γ k which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elements 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 Γ p ⊕ σ Γ p are isomorphic, but under the full Clifford algebra they are not.