Then splits in the ±"i" 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 Λ (as well as its complex conjugate Λ"V") 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. 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  In addition, for the real orthogonal groups, there are three characters (one-dimensional representations)  The Clebsch–Gordan decomposition allows one to define, among other things:  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 natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra: So also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose  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 of the representation Δ, and the following isomorphism is seen to hold:  In particular, note that the representation Δ of the orthochronous spin group is a unitary representation. In general, there are Clebsch–Gordan decompositions  In metric signature , the following isomorphisms hold for the conjugate half-spin representations Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the half-spin representations .