4.23.2 Tensor Argument Tuple Notation

Tensor Argument Tuple Notation is the set of conventions for abbreviating the full ordered list of a multilinear map's arguments, (v₁, ..., vₙ), as a single symbol, and for manipulating that symbol under permutation, substitution, and indexing without having to rewrite every individual component each time. This notation is what makes it practical to state general facts about arity-n multilinear maps without the visual clutter of writing out all n arguments explicitly at every step.

Abbreviating a Full Argument List

Bold or Vector-Style Tuple Symbols

A full tuple of arguments is frequently abbreviated by a single bold or overlined symbol, 𝐯 = (v₁, ..., vₙ), allowing a multilinear map to be written f(𝐯) in contexts where the individual components are not the focus of discussion, such as when comparing f(𝐯) to f(𝐰) for two entirely different tuples.

Distinguishing the Tuple From the Elementary Tensor

The tuple notation (v₁,...,vₙ), an element of the Cartesian product V₁ × ... × Vₙ, must be kept notationally distinct from the elementary tensor v₁ ⊗ ... ⊗ vₙ, an element of the tensor product V₁ ⊗ ... ⊗ Vₙ; the two are related by the canonical map ⊗, but they are elements of different spaces, and conflating their notation would obscure the distinction between a multilinear map's domain and the tensor product it factors through.

Notation for Permuted Tuples

Applying a Permutation to the Tuple

For a permutation σ ∈ Sₙ, the tuple with its entries rearranged is written 𝐯_σ = (v_{σ(1)}, ..., v_{σ(n)}), allowing the symmetric or alternating exchange rules to be stated compactly as f(𝐯_σ) = f(𝐯) (symmetric case) or f(𝐯_σ) = sgn(σ) f(𝐯) (alternating case), rather than writing out the full permuted argument list at each occurrence.

Transposition Notation for a Single Swap

The special case of swapping only two entries, positions i and j, is often denoted using cycle notation borrowed directly from permutation group theory, 𝐯_{(ij)}, denoting the tuple with entries i and j exchanged and all others unchanged, matching the notation used for the corresponding transposition in the symmetric group itself.

Multi-Index Notation

Bundling Several Indices Into One Symbol

When the entries of a tuple are themselves indexed, as in a component array T_{i₁...iₙ}, the full list of indices (i₁,...,iₙ) is frequently abbreviated as a single multi-index, often written in bold, 𝐈 = (i₁,...,iₙ), allowing the array to be written T_𝐈 and summation over all index values to be written ∑_𝐈, condensing an n-fold sum into a single summation symbol.

Multi-Index Arithmetic

Multi-index notation supports its own arithmetic conventions, such as |𝐈| = i₁ + ... + iₙ for the sum of the entries (used, for instance, in multivariate polynomial notation to denote total degree) and 𝐯^𝐈 = v^{i₁}⋯v^{iₙ} for a monomial indexed by the multi-index, both of which appear when translating between multilinear form component arrays and their associated homogeneous polynomials.

Notation for Substitution Into Slots

Substituting a Tuple Into a Fixed Position Pattern

When some argument slots of a multilinear map are to be filled from one tuple and others from another, subscripted position labels are used to specify which entries go where, such as f(𝐮_A, 𝐯_B) denoting that the entries of tuple 𝐮 fill the slot positions in a specified subset A while the entries of 𝐯 fill the complementary positions B; this notation is used when discussing partial symmetrization or when combining tuples from different sources into a single argument list.

Diagonal Substitution

Filling every slot of an n-ary multilinear map with copies of a single vector v is written f(v,...,v) or, using a diagonal-substitution notation, f(v^{(n)}), denoting the value of f restricted to the "diagonal" of V × ... × V; this notation appears whenever a symmetric multilinear form is being converted into its associated homogeneous polynomial p(v) = f(v,...,v).

Tuple Notation and Arity Changes

Concatenation of Tuples

Given a tuple 𝐮 = (u₁,...,uₘ) and a tuple 𝐯 = (v₁,...,vₖ), their concatenation 𝐮 · 𝐯 = (u₁,...,uₘ,v₁,...,vₖ) denotes the combined tuple of length m + k, used when describing the tensor product of two multilinear maps, (f ⊗ g)(𝐮 · 𝐯) = f(𝐮)g(𝐯), as a single map on the concatenated argument list rather than as two separate maps evaluated independently.

Truncated Tuple Notation for Currying

The tuple with its first entry removed, 𝐯' = (v₂,...,vₙ), is used when expressing the curried form of a multilinear map, f(v₁, 𝐯'), denoting that f is being viewed as a function of v₁ alone, returning a multilinear map applied to the remaining tuple 𝐯'.

Why Tuple Abbreviation Is Used

Reducing Visual Clutter in General Statements

Writing general facts about arity-n multilinear maps, the multilinearity condition, the symmetric or alternating exchange rules, the universal property's factorization equation, would require repeatedly writing out n explicit argument slots if tuple notation were not used; abbreviating the full argument list to a single symbol 𝐯 allows such statements to be written with the same brevity regardless of how large n happens to be.

Making the Arity-Independence of Certain Statements Visible

Because tuple notation hides the specific value of n, statements written using it, f(𝐯_σ) = sgn(σ)f(𝐯), for instance, are immediately seen to hold for every arity n simultaneously, rather than appearing as a family of separate statements, one for each arity, that happen to share a common pattern once the arguments are written out individually.