4.16.4 Tensor Extension Compatibility

Tensor Extension Compatibility is the condition that prescribed data on a spanning set of generators must satisfy in order for a well-defined multilinear extension to exist, requiring that the data respect every linear relation among the generators exactly as multilinearity itself would force. It is the precise criterion singled out within tensor multilinear extension construction that separates prescribed data admitting a genuine tensor extension from prescribed data that is simply inconsistent and extends to no tensor at all.

The Compatibility Condition in Detail

Relations Among Generators

Let g_1, ..., g_m span a vector space V, possibly with m > n where n = dim(V), so that nontrivial linear relations exist among the generators: there are scalars λ_1, ..., λ_m, not all zero, satisfying ∑_r λ_r g_r = 0. The collection of all such relations forms a subspace of F^m, called the relation space of the generating set.

The Condition Stated Precisely

Given prescribed data c(r_1, ..., r_{p+q}) for every combination of generator indices, compatibility requires that for every relation ∑_r λ_r g_r = 0 in the relation space, and for every slot l among the p + q slots,

where r occupies slot l and every other slot index is held fixed at some arbitrary combination. This must hold for every relation in the relation space and for every choice of the fixed indices in the remaining slots.

Why the Condition Is Necessary

What Multilinearity Would Force

If a genuine multilinear tensor T exists with T(..., g_r, ...) = c(..., r, ...) for every combination of generators, then substituting the relation ∑_r λ_r g_r = 0 into slot l and applying additivity and homogeneity of T in that slot gives ∑_r λ_r T(..., g_r, ...) = T(..., 0, ...) = 0, since a tensor evaluated with a zero argument in any slot vanishes; this is exactly the compatibility condition, so any tensor whose values happen to be read off on the generators automatically satisfies it.

What Failure of the Condition Reveals

If the compatibility condition fails for some relation and some slot, no multilinear map can have c as its values on the generators, since any hypothetical such map would be forced, by the argument above, to satisfy the condition; a failure therefore certifies, without any further work, that the prescribed data cannot arise from a genuine tensor.

Checking Compatibility in Practice

Reducing to a Basis of the Relation Space

Because the relation space is itself a vector space, it suffices to check the compatibility condition on a basis of the relation space rather than on every individual relation, since compatibility for a spanning set of relations, checked slot by slot, implies compatibility for every linear combination of those relations by linearity of the condition itself in the coefficients λ_r.

A Finite Verification Procedure

Since both the relation space and the number of slots are finite, verifying compatibility amounts to checking finitely many linear equations among the prescribed values c, one equation for each combination of a relation-space basis element and a slot, together with a choice of fixed indices for the remaining slots, making compatibility a decidable, mechanically checkable property of any given prescription.

Compatibility and the Basis Case

Vacuous Compatibility When No Relations Exist

When the generating set g_1, ..., g_m is linearly independent, meaning m = n and the relation space is the zero subspace, the compatibility condition is required to hold only for the trivial relation with every λ_r = 0, for which the condition reduces to 0 = 0; compatibility is therefore automatic in this case, matching the fact that tensor basis rule extension needs no separate consistency check.

Compatibility as a Genuine Restriction When Relations Exist

As soon as the generating set contains any nontrivial relation, compatibility becomes a real restriction on the admissible prescriptions of c, cutting down the space of otherwise arbitrary assignments to precisely those consistent with the linear structure already present among the g_r.

Consequences for the Space of Compatible Assignments

Compatible Assignments Form a Subspace

The set of all prescriptions c satisfying the compatibility condition forms a linear subspace of the full space of assignments to the generating set, since the compatibility condition is itself linear in c; this subspace is exactly the image of the restriction map sending tensors to their values on the generators, and its dimension equals n^{p+q}, the dimension of the space of tensors themselves, regardless of how large m is.

Redundancy Encoded in the Relation Space

The gap between the total number m^{p+q} of naively prescribable values and the dimension n^{p+q} of the compatible subspace reflects exactly the redundancy introduced by using more generators than a minimal basis would require, with the compatibility condition serving as the mechanism that removes this redundancy from the admissible data.

Diagrammatic Summary

The diagram shows the compatibility condition acting as a filter, selecting from all conceivable assignments on the generating set exactly those that could arise as the values of a genuine multilinear tensor.