4.18.2 Tensor Multilinear Form Argument Slots

Tensor Multilinear Form Argument Slots is the term for the individual input positions of a multilinear form, the n separate places into which vectors from V₁, ..., Vₙ are inserted, each of which the form treats linearly while every other slot is held fixed. Argument slots are the structural units that make multilinearity a condition checkable one position at a time, rather than a single global condition on the whole tuple of inputs.

What a Slot Is

Position, Not Just Value

A slot is a designated position i among the n inputs of f: V₁ × ... × Vₙ → W, associated to a specific factor space Vᵢ. Multilinearity is the requirement that, for each slot i separately, the function obtained by fixing every other slot and letting only the i-th vary is linear:

is a linear map Vᵢ → W, for every fixed choice of the other n - 1 vectors. Each slot therefore carries its own linearity condition, independent of the conditions imposed on the other slots.

Slots Are Independently Linear, Not Jointly Linear

A crucial distinction is that multilinearity does not mean the map f is linear as a function of the whole tuple (v₁, ..., vₙ) treated as a single vector in the direct sum V₁ ⊕ ... ⊕ Vₙ; in general f(v₁ + v₁', v₂, ..., vₙ) behaves linearly, but f(v₁ + v₁', v₂ + v₂', ..., vₙ) expands, for n ≥ 2, into a sum of 2ⁿ terms rather than simplifying to f(v₁,...,vₙ) + f(v₁',...,vₙ'). Argument slots make precise exactly what kind of linearity is present: linearity slot by slot, holding all others fixed, not linearity in the tuple as a whole.

Consequences of Fixing All Other Slots

Slot-wise Linear Maps

For fixed vectors v₁, ..., v_{i-1}, v_{i+1}, ..., vₙ, the induced map on slot i alone is a genuine linear map Vᵢ → W, and can be treated with all the standard tools of linear algebra: it has a kernel, a rank, a matrix representation relative to a basis of Vᵢ, and so on, even though f as a whole is not linear.

Partial Application and Currying

Fixing the first k slots of f produces a multilinear map of the remaining n - k slots:

and this construction can be iterated slot by slot until a single vector remains in a single argument. This gives a currying correspondence between an n-ary multilinear map and a linear map into a space of (n-1)-ary multilinear maps, expressed as

reducing the study of an n-slot multilinear map to a linear map whose target is itself a space of maps with one fewer slot.

Slots and the Correspondence With Elementary Tensors

Slot Order Matches Tensor Factor Order

The n argument slots of a multilinear map correspond, under the universal property of the tensor product, to the n tensor factors of V₁ ⊗ ... ⊗ Vₙ. Filling slot i with the vector vᵢ corresponds to placing vᵢ in the i-th position of the elementary tensor v₁ ⊗ ... ⊗ vₙ; the order of the slots is therefore not incidental notation but is exactly mirrored by the order of factors in the tensor product.

Permuting Slots

Permuting the argument slots of a multilinear form, that is, considering f(v_{σ(1)}, ..., v_{σ(n)}) for a permutation σ, produces a new multilinear form, and this operation corresponds on the tensor side to the linear automorphism of V ⊗ ... ⊗ V (when all factors coincide) that permutes tensor factors according to σ. A form is called symmetric if it is unchanged by every such permutation of its slots, and alternating if it changes sign under every transposition of two slots, both notions defined purely in terms of how the form behaves when its argument slots are permuted.

Slots in Coordinate Computations

Index Assigned per Slot

When each Vᵢ has a chosen basis, the coordinate array T_{k₁...kₙ} representing a multilinear form assigns one index to each argument slot, so that k_i ranges over the basis of Vᵢ, the space associated to slot i. Operations that act on a single slot, such as raising or lowering an index using a metric, or contracting a slot against a dual vector, are described entirely in terms of the index associated to that slot, leaving the indices of other slots untouched.

Slot-Restricted Linear Operations

Given a linear map φ: Vᵢ → Vᵢ' acting only on the space associated to slot i, precomposing f with φ in that slot alone, that is, forming f(v₁, ..., φ(vᵢ'), ..., vₙ) as a function of vᵢ' instead of vᵢ, produces a new multilinear form on V₁, ..., Vᵢ', ..., Vₙ. On the tensor side, this corresponds to applying id ⊗ ... ⊗ φ ⊗ ... ⊗ id to the corresponding tensor, acting on only the i-th tensor factor while leaving the others fixed, which is the direct tensor-level counterpart of modifying a single argument slot.

Why the Slot Perspective Matters

Localizing Verification

Because multilinearity is a per-slot condition, verifying that a candidate map is multilinear reduces to n separate, individually simple linearity checks, one for each slot, rather than a single more complicated joint condition; this is the practical reason the definition is stated slot by slot rather than as a single global linearity requirement over the whole tuple.

Localizing Modification

The slot perspective also clarifies how multilinear forms can be modified locally: composing with a map, restricting to a subspace, or fixing a value can be done independently in each slot without disturbing the linearity already established in the other slots, which is what allows constructions such as tensor products of forms, contractions, and symmetrizations to be built up slot by slot rather than requiring the whole form to be redefined from scratch.