4.3.2 Tensor Trilinear Second Argument Slot

The Tensor Trilinear Second Argument Slot is the second position in a trilinear map where a tensor operand is applied.

Tensor Trilinear Second Argument Slot is the designated input position of a trilinear map that, when the first and third arguments are both held fixed at any chosen values, receives a linear function of that single remaining variable. Given a trilinear map $T : V \times W \times X \to U$ the second argument slot is the coordinate reserved for elements of $W$ , and trilinearity guarantees that fixing any $v \in V$ and $x \in X$ produces a linear map $T (v, \cdot, x) : W \to U$ acting purely on the second slot.

Formal Statement

The Linearity Condition

For every fixed pair $v \in V$ , $x \in X$ , every $w_{1}, w_{2} \in W$ , and every scalar $β$ , the second argument slot satisfies

T (v, w_{1} + β w_{2}, x) = T (v, w_{1}, x) + β T (v, w_{2}, x)

As with every slot of a trilinear map, this linearity holds only once both other arguments are frozen; the map is not, in general, linear jointly in the second slot together with either of the others.

Notation for the Frozen Partial Map

The partial map obtained by freezing the first and third slots is written

T_{v, x} : W \to U, T_{v, x} (w) = T (v, w, x)

Coordinate Description

Index Position in the Structure Constants

Relative to bases with structure constants $c_{i j k}^{l}$ , the second argument slot corresponds to the index $j$ . Fixing $i$ and $k$ leaves an expression linear in $w^{j}$ :

T (v, w, x) = \sum_{j} w^{j} (\sum_{i, k} c_{i j k}^{l} v^{i} x^{k}) u_{l}

The middle position of the index $j$ in the array reflects the second slot's position between the first and third in the ordered argument list, a bookkeeping detail that becomes essential when the tensor lacks symmetry among its indices.

Distinction from the First and Third Slots

Why Middle Position Matters

Unless a trilinear map is fully symmetric, the second slot is not interchangeable with the first or third, and swapping the second argument with either neighbor generally changes the value of the map. For example, in a trilinear map built from the composition of two bilinear maps, the second slot's index often plays the role of an intermediate "bridge" between the maps, giving it structural significance distinct from the endpoint slots.

Effect of Partial Symmetry

Some trilinear maps are symmetric only between the first and third slots while treating the second differently, as occurs in the coefficients of certain connection or curvature-related tensors, where the middle index is singled out by the definition of the tensor rather than by an arbitrary choice, making explicit identification of the second argument slot necessary for correct interpretation.

Currying and Contraction

Embedding into the Middle Tensor Factor

Under the universal property of the triple tensor product, the second argument slot's linearity is exactly what allows the assignment $w \mapsto v \otimes w \otimes x$ , for fixed $v$ and $x$ , to be linear, identifying the second slot with the middle tensor factor in $V \otimes W \otimes X$ .

Contraction on the Middle Index

Contracting a rank-three tensor against a covector in its second index produces a rank-two tensor by evaluating the second argument slot at a fixed vector while leaving the first and third slots free; because tensor contraction is defined index by index, correctly identifying the second slot is required to contract the intended index rather than the first or third.

Summary of Structural Role

The second argument slot completes, together with the first and third slots, the full linear decomposition of a trilinear map into three independent one-variable linear maps once the complementary pair of arguments is fixed; its position in the middle of the argument list gives it a distinguished role whenever the underlying tensor exhibits only partial symmetry among its three indices.