15.3.4 Tensor Symmetric Type Requirement

Tensor Symmetric Type Requirement is the condition on a tensor's valence, meaning the arrangement of its upper and lower indices, that must be satisfied before symmetrization of a group of indices is even a well-defined operation. It specifies that symmetry can only be imposed among indices that share the same variance type, since attempting to symmetrize indices of different types mixes objects that do not transform the same way under a change of basis.

Same-Variance Requirement

Why Variance Type Matters

A tensor of type ( (p, q) ) has ( p ) upper (contravariant) indices and ( q ) lower (covariant) indices, and these two kinds of indices transform under a basis change using inverse matrices of one another. Because symmetrization is defined as an average over permutations of index positions, permuting an upper index into a slot occupied by a lower index would require comparing quantities that transform differently, which is not a coordinate-independent operation.

Stating the Requirement

The type requirement is stated as: symmetrization is permitted only among a subset of upper indices, or only among a subset of lower indices, but never across the two:

is well defined, since ( i ) and ( k ) are both upper indices, while an expression that attempts to symmetrize an upper index together with a lower index does not correspond to any valid tensor operation without additional structure being introduced first.

Role of a Metric in Relaxing the Requirement

Raising and Lowering Before Symmetrization

When a metric tensor is available, it can convert a lower index into an upper one, or an upper index into a lower one, prior to symmetrization. This does not violate the type requirement; instead, it satisfies the requirement first by bringing every index to be symmetrized into the same variance type, and only then performs the symmetrization:

where the metric ( g ) has been used to raise the lower index ( k ) so that the resulting expression symmetrizes two indices of the same type.

Without a Metric, the Requirement Is Strict

In the absence of a metric or any other structure capable of converting index type, the type requirement remains strict: an upper and a lower index simply cannot be symmetrized together, and any notation that appears to do so is either an abuse of notation or implicitly assumes a metric has already been applied.

Requirement Under Change of Basis

Preservation of the Requirement

The type requirement is preserved under any admissible change of basis: since upper indices continue to transform contravariantly and lower indices continue to transform covariantly in every basis, a symmetrization that respects the type requirement in one basis automatically respects it in every other basis reached by a linear transformation.

Consequence for Tensor Classification

Because of this preservation, the type requirement is not merely a notational convenience but a structural fact about the tensor: a tensor described as symmetric in a given set of upper indices, or in a given set of lower indices, retains that classification regardless of which coordinate system is used to express its components.

Interaction With Rank and Multiple Index Groups

Independent Symmetry Within Each Group

A tensor of type ( (p, q) ) can carry independent symmetry requirements for its upper indices and for its lower indices separately, since each group satisfies the same-variance requirement on its own. A tensor can therefore be symmetric among its ( p ) upper indices while carrying no particular symmetry among its ( q ) lower indices, or vice versa, without any contradiction.

Consequence for the Symmetric Rank Area

When the type requirement is satisfied, the counting formula for the symmetric rank area applies separately to the upper index group and to the lower index group, and the total number of independent components of the full mixed tensor is the product of the two group-specific symmetric rank areas, since the two groups vary independently of one another under the type requirement.