6.10.2 Tensor Mixed Contravariant Slot Set

Tensor Mixed Contravariant Slot Set is the collection, considered as a distinguishable group, of every upper-index argument position belonging to a mixed type tensor, set apart from the tensor's lower-index positions, so that the one-form-accepting slots of a tensor carrying both variances can be examined, ordered, and manipulated as their own coherent subset rather than being treated as scattered individually among the tensor's full list of indices. Where a purely contravariant tensor has only one kind of slot to speak of, a mixed type tensor has two distinct kinds coexisting on the same object, and the mixed contravariant slot set names exactly the sub-collection made up of the one-form-accepting kind.

Isolating the Contravariant Slots Within a Mixed Tensor

Partitioning the Full Slot List

A mixed type tensor of contravariant order p and covariant order q has p plus q slots in total, but these do not form a single undifferentiated list: they split naturally into two groups, the p slots that accept one-forms and the q slots that accept vectors. The mixed contravariant slot set is precisely this first group, considered on its own, independently of whatever lower slots happen to sit alongside it on the same tensor.

Internal Order Within the Set

The slots making up the mixed contravariant slot set retain their own internal ordering relative to one another, corresponding to the left-to-right sequence in which the tensor's upper indices are written, even though this ordering is entirely separate from wherever the lower indices happen to be interleaved in the full notation. A tensor with two upper indices possesses a contravariant slot set of size two whose first and second members are as distinguishable from each other as they would be in a purely contravariant tensor of the same order.

Behavior of the Contravariant Slot Set Under Operations

Filling the Set Independently of the Lower Slots

Supplying one-forms to every member of the mixed contravariant slot set while leaving the tensor's lower slots unfilled produces a partially evaluated object whose remaining open positions are exactly the original lower slots. The reverse also holds: filling every lower slot with vectors while leaving the contravariant slot set untouched produces an object whose remaining open positions are exactly the original contravariant slot set. The two groups of slots can therefore be evaluated in either order, or interleaved, without the final scalar depending on the order in which the two sets were saturated, since multilinearity guarantees this independence.

Symmetrization and Antisymmetrization Confined to the Set

Symmetrizing or antisymmetrizing a mixed type tensor over two of its upper indices operates strictly within the mixed contravariant slot set, exchanging or averaging the one-forms assigned to two members of that set while leaving every lower slot completely untouched. This confinement is essential: a mixed type tensor can be fully symmetric among the members of its contravariant slot set while showing no particular symmetry among its lower slots, or vice versa, because permutation operations native to one variance never reach across into the other.

Contraction Removing a Member of the Set

Contracting one particular upper index of a mixed type tensor against a lower index, whether that lower index belongs to the same tensor or to a separate one being multiplied against it, removes exactly one member from the mixed contravariant slot set, leaving a smaller set of size one less than before. The remaining members of the set retain their relative order exactly as they stood prior to the contraction, with only the contracted position excised.

Significance of Treating the Contravariant Slots as a Set

Enabling Type Reduction Statements

Describing the mixed contravariant slot set as its own entity makes it possible to state precisely how a mixed type tensor reduces to a purely covariant tensor: saturating every member of the contravariant slot set with fixed one-forms removes the entire set at once, leaving behind an object governed solely by the original lower slots and hence purely covariant in character.

Clarifying What the Metric Can and Cannot Change

Lowering one member of the mixed contravariant slot set with the metric transfers that particular slot out of the contravariant slot set and into the collection of lower slots, shrinking the contravariant slot set by one member while growing the lower slot collection by one. Framing this operation in terms of the sets involved makes clear that the total number of slots on the tensor is unchanged by raising or lowering; only the partition of slots between the two sets shifts.

Distinguishing the Set From the Slot Count Alone

The mixed contravariant slot set carries more information than the contravariant order taken as a bare number, since the set also records the identity and relative position of each member slot, not merely how many there are. Two mixed type tensors can share an identical contravariant order while having contravariant slot sets that behave completely differently under symmetrization, because the number of members being equal says nothing about the symmetry relations holding among those particular members.