4.14.2 Tensor Second Slot Homogeneity

Tensor Second Slot Homogeneity is the specific instance of the tensor multilinear homogeneity property applied to the second argument position of a multilinear map, requiring that scaling the second argument by a constant, while every other argument is held fixed, produces an output scaled by exactly that same constant. It is the counterpart to tensor second slot additivity, and together the two conditions establish that the second slot of a tensor behaves as an ordinary linear map whenever the other slots are held fixed.

Statement for the Second Slot

The Defining Identity

For a type (p, q) tensor T on a vector space V over a field F with at least two argument slots, second slot homogeneity states that for any admissible argument u in the second position, any scalar λ ∈ F, and any fixed choice of every other argument,

where x_1 denotes the first argument and x_3, ... denote every argument from the third slot onward, all held fixed and identical on both sides. Only the second slot is scaled, by λ on the left, and the entire output is scaled by that same λ on the right.

The Kind of Object Filling the Second Slot

Whether the second slot is contravariant or covariant is determined by the type of T; second slot homogeneity is stated using whichever kind of argument, covector or vector, that slot expects, and the scalar λ acts by the ordinary scalar multiplication defined on V or V* respectively.

Illustration in the Bilinear Case

The Simplest Setting: Two Arguments

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

with the first argument x held fixed throughout, showing that B responds proportionally to any rescaling of its second argument, for every fixed choice of the first.

Contrast with First Slot Homogeneity

First slot homogeneity for the same bilinear form B instead requires B(λx, u) = λ B(x, u), with the second argument fixed; first slot homogeneity and second slot homogeneity are logically independent conditions, and a bilinear form must satisfy both simultaneously, since multilinearity requires homogeneity in every slot without exception.

Second Slot Homogeneity 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, second slot homogeneity and first slot homogeneity become logically equivalent statements about that tensor, since the symmetry relation directly translates the scaling behavior in one slot into the scaling behavior 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, second slot homogeneity again follows from first slot homogeneity, and vice versa, since scaling one side of the antisymmetry relation and comparing coefficients reproduces the homogeneity identity for the other slot.

Consequences Restricted to the Second Slot

Vanishing When the Second Argument Is Zero

Setting λ = 0 in second slot homogeneity shows that T(x_1, 0, x_3, ...) = 0 for any fixed choice of the remaining arguments, meaning the tensor automatically vanishes whenever its second argument is the zero vector or zero covector, regardless of what values occupy the other slots.

Second Slot Homogeneity Alone Is Insufficient for Multilinearity

Establishing that a candidate map satisfies second slot homogeneity says nothing, by itself, about its behavior in the first, third, or any other slot; a map could be homogeneous in its second argument while failing homogeneity in a different argument, and such a map would not be classified as fully multilinear despite satisfying second slot homogeneity in isolation.

Its Place in a Full Multilinearity Check

Paired with Second Slot Additivity

Confirming that the second slot of a tensor behaves linearly requires checking both second slot additivity and second slot homogeneity together; neither condition alone establishes linearity in the second slot, since a map could satisfy one without the other.

One Step Among Many

Second slot homogeneity is one of p + q homogeneity checks needed to establish full multilinearity of a tensor, one for each slot; it must be supplemented by identical homogeneity checks, alongside additivity checks, for every remaining slot before multilinearity can be concluded.

Diagrammatic Summary

The diagram illustrates second slot homogeneity by showing the first argument x held fixed while scaling the second argument by λ produces an output scaled by that same λ.