6.10.1 Tensor Mixed Covariant Slot Set

Tensor Mixed Covariant Slot Set is the collection, considered as a distinguishable group, of every lower-index argument position belonging to a mixed type tensor, set apart from the tensor's upper-index positions, so that the vector-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 covariant 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 covariant slot set names exactly the sub-collection made up of the vector-accepting kind.

Isolating the Covariant 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 covariant slot set is precisely this second group, considered on its own, independently of whatever upper slots happen to sit alongside it on the same tensor.

Internal Order Within the Set

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

Behavior of the Covariant Slot Set Under Operations

Filling the Set Independently of the Upper Slots

Supplying vectors to every member of the mixed covariant slot set while leaving the tensor's upper slots unfilled produces a partially evaluated object whose remaining open positions are exactly the original upper slots. The reverse also holds: filling every upper slot with one-forms while leaving the covariant slot set untouched produces an object whose remaining open positions are exactly the original covariant 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 lower indices operates strictly within the mixed covariant slot set, exchanging or averaging the vectors assigned to two members of that set while leaving every upper slot completely untouched. This confinement is essential: a mixed type tensor can be fully symmetric among the members of its covariant slot set while showing no particular symmetry among its upper 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 lower index of a mixed type tensor against an upper index, whether that upper index belongs to the same tensor or to a separate one being multiplied against it, removes exactly one member from the mixed covariant 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 Covariant Slots as a Set

Enabling Type Reduction Statements

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

Clarifying What the Metric Can and Cannot Change

Raising one member of the mixed covariant slot set with the inverse metric transfers that particular slot out of the covariant slot set and into the collection of upper slots, shrinking the covariant slot set by one member while growing the upper 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 covariant slot set carries more information than the covariant 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 covariant order while having covariant 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.