4.13.2 Tensor Second Slot Additivity

Tensor Second Slot Additivity is the specific instance of the tensor multilinear additivity property applied to the second argument position of a multilinear map, requiring that replacing the second 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 serves as the concrete, singled-out example most often used to illustrate additivity in a particular slot, since a second slot is the simplest position at which the general single-slot pattern can be shown alongside a first slot without ambiguity about which argument is varying.

Statement for the Second Slot

The Defining Identity

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

where x_1 denotes the first argument, held fixed and identical on both sides, and x_3, ... denote every argument from the third slot onward, also held fixed and identical on both sides. Only the second slot is permitted to vary, receiving u + w on the left and u, then w, separately on the right.

Whether the Slot Is Contravariant or Covariant

The kind of object admissible in the second slot, a covector if that slot is contravariant or a vector if it is covariant, is fixed by the type of T; second slot additivity is stated using whichever kind of argument that slot expects, and the identity above holds regardless of whether the second slot happens to be contravariant or covariant, since additivity is a property shared by every slot uniformly.

Illustration in the Bilinear Case

The Simplest Setting: Two Arguments

The clearest illustration of second slot additivity occurs for a type (0, 2) tensor B, a bilinear form taking two vector arguments. Second slot additivity for B reads

with the first argument x held fixed throughout, showing that B is additive in its second argument for every choice of the first, exactly analogous to first slot additivity but with the roles of the two arguments reversed.

Contrast with First Slot Additivity

First slot additivity for the same bilinear form B requires B(x + y, w) = B(x, w) + B(y, w), with the second argument fixed instead; the two properties, first slot and second slot additivity, are logically independent statements about B, and a bilinear form is required to satisfy both simultaneously, since multilinearity demands additivity in every slot, not merely in one.

Second Slot Additivity and Symmetric or Antisymmetric Tensors

Automatic Transfer Under Symmetry

If a tensor is symmetric in its first two slots, meaning swapping the first and second arguments leaves the output unchanged, then first slot additivity and second slot additivity become equivalent statements about the same tensor, since the symmetry relates the behavior in one slot directly to the behavior in the other.

Automatic Transfer Under Antisymmetry

Similarly, if a tensor is antisymmetric in its first two slots, so that swapping the first and second arguments negates the output, second slot additivity again follows directly from first slot additivity, since negating both sides of the first slot additivity identity and applying the antisymmetry relation reproduces the second slot additivity identity.

Consequences Restricted to the Second Slot

Second Slot Additivity Alone Does Not Imply Full Multilinearity

Establishing that a map is additive in its second slot says nothing, by itself, about its behavior in the first, third, or any other slot; a map could be additive in the second slot while failing additivity in the first slot, and such a map would not qualify as fully multilinear, even though it satisfies second slot additivity in isolation.

Role in Verifying Multilinearity Slot by Slot

In practice, confirming that a candidate map is multilinear typically proceeds by checking additivity, together with homogeneity, in each slot one at a time; second slot additivity is one specific check in this sequence, verified independently of, and in addition to, the corresponding checks for every other slot of the map.

Diagrammatic Summary

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