11.18.3 Tensor Variance Type Mixed Arrangement

Tensor Variance Type Mixed Arrangement is the specific ordering and slot structure given to the upper and lower indices of a mixed tensor, determining not merely how many contravariant and covariant indices are present but exactly which position each individual index occupies relative to the others, a detail that matters whenever the tensor is contracted, differentiated, or compared with another tensor of the same type.

Foundational Setting

Beyond Counting Indices

Knowing that a tensor has variance type $$(p,q)$$ fixes how many upper and lower indices it carries, but it does not by itself specify the arrangement of those indices. Two tensors can share the same type yet differ in mixed arrangement if the relative ordering of their indices, either among themselves or between upper and lower groups, differs.

Grouped Versus Interleaved Notation

A common arrangement groups all upper indices together, followed by all lower indices, as in $$T_{kl}^{ij}$$. An alternative interleaved arrangement places indices in the order they logically arise in a formula, as in $$T_k^i$$ followed by a separate factor with its own indices; both arrangements describe valid tensors, but the specific slot each index occupies must be tracked consistently.

Why Slot Order Matters

Contraction Depends on Matching Slots

When contracting a specific upper index of one tensor against a specific lower index of another, the mixed arrangement determines exactly which pair of indices is summed. Given $$S_i^j$$ and $$T_k^l$$, contracting the second tensor's lower index against the first tensor's upper index produces a result distinct from contracting against a different slot, even if both slots are nominally lower indices.

Symmetry Properties Depend on Position

Whether a tensor is symmetric or antisymmetric in a pair of indices is only meaningful when those indices occupy slots of the same type and are compared under an explicit swap. A tensor symmetric in two upper indices satisfies:

but no such comparison is directly meaningful between an upper index slot and a lower index slot, since they follow different transformation rules.

Visualizing Slot Structure

Diagram of Index Slots

Arrangement in Standard Constructions

The Identity Operator's Arrangement

The Kronecker delta, viewed as the mixed tensor representing the identity linear map, has a canonical single-upper, single-lower arrangement, $${\delta }_j^i$$, and swapping which index is upper and which is lower would describe a different, generally inequivalent object unless the space carries additional structure identifying the two slots.

Curvature and Multi-Slot Tensors

Tensors describing curvature typically carry a specific, conventionally fixed arrangement of several lower indices and one upper index, and different textbooks may adopt different but equivalent orderings of these slots, making explicit awareness of the mixed arrangement essential when comparing formulas across sources.

Arrangement Under Tensor Product

Preserving Relative Slot Order

When two tensors are combined by tensor product, the mixed arrangement of the result places all indices of the first factor before those of the second factor, in their original relative order, unless indices are explicitly relabeled or permuted, so the arrangement of a product tensor is fully determined by the arrangements of its factors.

Relabeling Does Not Change the Tensor

Renaming a dummy or free index letter, without altering which slot it occupies or whether it is upper or lower, leaves the tensor and its mixed arrangement unchanged, since the letter itself carries no independent meaning beyond marking a specific slot.

Summary of Key Traits

Defining Characteristics

• Mixed arrangement specifies the exact slot position of each index, beyond merely counting upper and lower indices.
• Contraction and symmetry statements depend on which specific slots are being compared or summed.
• Standard tensors such as the Kronecker delta and curvature tensors follow conventional, fixed arrangements.
• Tensor products preserve the relative slot order of their factors, and relabeling index letters does not alter the underlying arrangement.