the case that the real vector space V is a hermitian vector space , i.e., V is equipped with a complex structure J that is an orthogonal transformation with respect to the inner product g on V. Then splits in the eigenspaces of J. These eigenspaces are isotropic for the complexification of g and can be identified with the complex vector space and its complex conjugate . Therefore, for a hermitian vector space the vector space (as well as its complex conjugate is a spinor space for the underlying real euclidean vector space. With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an almost Hermitian manifold and is the reason why every almost complex manifold (in particular every symplectic manifold) has a Spinc structure. Likewise, every complex vector bundle on a manifold carries a Spinc structure. Clebsch–Gordan decomposition A number of Clebsch–Gordan decompositions are possible on the tensor product of one spin representation with another. These decompositions express the tensor product in terms of the alternating representations of the orthogonal group. For the real or complex case, the alternating representations are , the representation of the orthogonal group on skew tensors of rank r. In addition, for the real orthogonal groups, there are three characters (one-dimensional representations) σ+ : O(p, q) → {−1, +1} given by , if R reverses the spatial orientation of V, +1, if R preserves the spatial orientation of V. (The spatial character.) σ− : O(p, q) → {−1, +1} given by , if R reverses the temporal orientation of V, +1, if R preserves the temporal orientation of V. (The temporal character.) σ = σ+σ− . (The orientation character.) The Clebsch–Gordan decomposition allows one to define, among other things: An action of spinors on vectors. A Hermitian metric on the complex representations of the real spin groups. A Dirac operator on each spin representation. Even dimensions If is even, then the tensor product of Δ with the contragredient representation decomposes as which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elements . 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 are isomorphic, but under the full Clifford algebra they are not. There is a