4.13.1 Tensor First Slot Additivity

Tensor First Slot Additivity is the specific instance of the tensor multilinear additivity property applied to the first argument position of a multilinear map, requiring that replacing the first argument with a sum of two vectors or covectors, while every other argument is held fixed, produces an output equal to the sum of the two outputs obtained from each summand individually. It is the natural starting point among the slot-by-slot additivity conditions, since the first slot is the position most commonly varied first when verifying, or applying, the multilinearity of a tensor.

Statement for the First Slot

The Defining Identity

For a type (p, q) tensor T on a vector space V, first slot additivity states that for any two admissible arguments u and w in the first position, and any fixed choice of every remaining argument,

where x_2, ... denotes every argument from the second slot onward, held fixed and identical on both sides. Only the first slot is permitted to vary, receiving u + w on the left and u, then w, separately on the right.

The Kind of Object Filling the First Slot

Whether the first slot is contravariant or covariant is determined by the type of T; if T has type (p, q) with p > 0, the first slot is conventionally taken to be contravariant, expecting a covector, while if p = 0 the first slot is covariant, expecting a vector. First slot additivity is stated using whichever kind of argument the first slot actually expects, according to the type of T.

Illustration in the Bilinear Case

The Simplest Setting: Two Arguments

For a type (0, 2) tensor B, a bilinear form taking two vector arguments, first slot additivity reads

with the second argument y held fixed throughout, showing that B is additive in its first argument for every choice of the second.

Contrast with Second Slot Additivity

Second slot additivity for the same bilinear form B instead requires B(x, u + w) = B(x, u) + B(x, w), with the first argument fixed; first slot additivity and second slot additivity are logically independent conditions, and a bilinear form must satisfy both, since multilinearity requires additivity in every slot simultaneously, not merely in one chosen slot.

First Slot Additivity and Symmetry Relations

Transfer Under Symmetric Tensors

If a tensor is symmetric in its first two slots, so that swapping the first and second arguments leaves the output unchanged, then first slot additivity and second slot additivity become logically equivalent statements about that tensor, since the symmetry relation directly translates the additivity condition in one slot into the additivity condition in the other.

Transfer Under Antisymmetric Tensors

If a tensor is antisymmetric in its first two slots, so that swapping the first and second arguments negates the output, first slot additivity again implies second slot additivity, and vice versa, since negating both sides of one additivity identity and applying the antisymmetry relation reproduces the additivity identity for the other slot.

First Slot Additivity Alone Is Insufficient for Multilinearity

No Automatic Guarantee for Other Slots

Verifying that a candidate map satisfies first slot additivity establishes nothing, by itself, about the behavior of that map in its second, third, or any later slot; a map could be additive in its first argument while failing additivity in a later argument, and such a map would not be classified as fully multilinear despite satisfying first slot additivity in isolation.

Its Place in a Full Multilinearity Check

Confirming full multilinearity of a candidate tensor requires checking additivity, together with homogeneity, separately in every one of its p + q slots; first slot additivity is simply the first of these checks in a natural left-to-right ordering of the slots, and it must be supplemented by identical checks for every remaining slot before multilinearity can be concluded.

Diagrammatic Summary

The diagram illustrates first slot additivity by showing the second argument y held fixed while the first argument, split into a sum u + w, produces the same output as adding the two separate evaluations on u and on w.