1.2.23 Higher Multilinear Map Definition

Higher Multilinear Map Definition is the characterization of a higher multilinear map as a multilinear map taking four or more vector arguments, drawn from one or more vector spaces defined over a common scalar field, that is linear in each argument separately whenever every other argument is held fixed. It extends the pattern already established by the linear map, the bilinear map, and the trilinear map to arbitrary finite arity, and it is the general form ultimately needed to accommodate tensors of arbitrarily high rank within the formal definition of a tensor.

Extending the Pattern Beyond Three Arguments

Once the definitions of the bilinear map and the trilinear map have established the pattern of requiring linearity independently in each of two, and then three, argument positions, the extension to four, five, or any larger finite number of arguments follows the same template without introducing any new conceptual ingredient: a function of k vector arguments is multilinear of degree k, or a higher multilinear map, if it is linear in each of its k arguments whenever the remaining k minus one arguments are held fixed. What changes as the number of arguments grows is not the nature of the linearity condition, but the combinatorial richness of the properties, such as symmetry, that the map may exhibit across its many arguments.

The expression above states the general defining requirement of a higher multilinear map of arity k: linearity must hold separately in every one of the k argument positions, with the remaining positions held fixed in each case.

Symmetry Classes Among Many Arguments

As the number of arguments grows, the possible symmetry patterns a multilinear form may exhibit multiply considerably. A fully symmetric multilinear form of arity k is unchanged under any of the many possible permutations of its k arguments, and a fully alternating multilinear form changes sign under every transposition of two arguments among the k. Between these two extremes lie multilinear maps symmetric or alternating only with respect to some subset of their arguments, or exhibiting more intricate symmetry patterns describable using the representation theory of the permutation group acting on the k argument positions. This growing combinatorial complexity is one of the principal reasons higher-rank tensors, corresponding to higher multilinear maps, support a correspondingly richer classification than tensors of low rank.

Higher Multilinear Maps and Tensor Rank

Every higher multilinear map, regardless of its arity, is subject to the same universal property that governs bilinear and trilinear maps: it corresponds to a unique linear map on the tensor product of the vector spaces from which its arguments are drawn. A higher multilinear map of arity k, taking arguments from combinations of a vector space and its dual, corresponds precisely to a tensor of total rank k, with the specific type of the tensor — how many of its indices are covariant and how many are contravariant — determined by how many of the k arguments are drawn from the dual space and how many from the original space.

Why the General Case Matters

Restricting attention only to linear, bilinear, and trilinear maps would artificially cap the rank of tensors that could be defined at three, which is far too limited for the applications tensor algebra is designed to support: the Riemann curvature tensor of general relativity, for instance, is a rank-four tensor, and elasticity tensors describing the response of anisotropic materials can reach rank four as well. Stating the definition of a multilinear map in full generality, for an arbitrary finite number of arguments, is therefore not a mere formal nicety but a practical necessity, ensuring that the tensor concept, once defined, places no artificial ceiling on the rank of the objects it can describe.