unavoidable ambiguity with such notation depending on whether one views as multilinear maps or as linear maps in which case the corresponding norms and inner products would differ by a constant factor. Although this would not lead to any inconsistencies in the above equations, since all terms would be changed by the same factor, it can lead to confusion in more involved contexts. For this reason, the index notation can often be easier to understand. Related formulas One can compute the "norm formulas" and the "trace formulas" Mathematical explanation of the decomposition Mathematically, the Ricci decomposition is the decomposition of the space of all tensors having the symmetries of the Riemann tensor into its irreducible representations for the action of the orthogonal group . Let V be an n-dimensional vector space, equipped with a metric tensor (of possibly mixed signature). Here V is modeled on the cotangent space at a point, so that a curvature tensor R (with all indices lowered) is an element of the tensor product V⊗V⊗V⊗V. The curvature tensor is skew symmetric in its first and last two entries: and obeys the interchange symmetry for all x,y,z,w ∈ V∗. As a result, R is an element of the subspace , the second symmetric power of the second exterior power of V. A curvature tensor must also satisfy the Bianchi identity, meaning that it is in the kernel of the linear map given by The space in S2Λ2V is the space of algebraic curvature tensors. The Ricci decomposition is the decomposition of this space into irreducible factors. The Ricci contraction mapping is given by This associates a symmetric 2-form to an algebraic curvature tensor. Conversely, given a pair of symmetric 2-forms h and k, the Kulkarni–Nomizu product of h and k produces an algebraic curvature tensor. If n > 4, then there is an orthogonal decomposition into (unique) irreducible subspaces where , where is the space of real scalars , where SV is the space of trace-free symmetric 2-forms The parts S, E, and C of the Ricci decomposition of a given Riemann tensor R are the orthogonal projections of R onto these invariant factors. In particular, is an orthogonal decomposition in the sense that This decomposition expresses the space of tensors with Riemann symmetries as a direct sum of the scalar submodule, the Ricci submodule, and Weyl submodule, respectively. Each of these modules is an irreducible representation