1.3.4 Tensor Type Structure

Tensor Type Structure defines the formal framework for classifying tensors by their transformation properties, essential in algebraic and geometric applications.

Tensor Type Structure is the organizing framework by which tensors are classified according to the ordered pair (p, q), distinguishing not merely how many indices a tensor carries in total but precisely how those indices are distributed between contravariant, upper, positions and covariant, lower, positions. Where order structure tracks only the sum p + q, type structure tracks the full pair, arranging all tensor spaces built from a given vector space V into a two-dimensional lattice indexed by p along one axis and q along the other, with each lattice point T^p_q(V) a distinct space in its own right.

The Type Lattice

Two Independent Parameters

Every tensor space T^p_q(V) is specified by two independent nonnegative integers, p, the number of contravariant indices, and q, the number of covariant indices. Unlike order, which collapses this pair into a single number p + q, type structure preserves the distinction, recognizing that a type (2, 0) tensor and a type (1, 1) tensor, though both order two, occupy different positions in the lattice and behave differently under a change of basis.

T_{q}^{p} (V) = \otimes_{i = 1}^{p} V \otimes \otimes_{j = 1}^{q} V^{*}

Lattice Coordinates

Each type (p, q) can be plotted as a lattice point on a grid, with p increasing along one axis and q increasing along the other. Type (0, 0), the scalars, sits at the origin of this lattice, and every other type is reached by moving some number of steps in the p direction and some number of steps in the q direction.

Duality Within the Type Structure

The Duality Between p and q

The two families T^p_0(V) and T^0_q(V), formed by taking p copies of V alone or q copies of V* alone, are dual to one another under the natural pairing extended from the pairing of V and V*: an element of T^p_0(V) can be evaluated on an element of T^0_p(V) to produce a scalar, mirroring the basic duality between vectors and covectors from which the entire type lattice is built.

T_{0}^{p} (V) ≅ {(T_{p}^{0}, (V))}^{*}

Symmetry of the Lattice Under Exchange

The lattice of tensor types exhibits a structural symmetry under exchanging p and q: the space T^p_q(V) and the space T^q_p(V) have the same dimension, n^(p+q), and are related by systematically swapping the roles of V and V*, though they remain distinct spaces unless further structure, such as a metric, is introduced to identify V with V*.

Movement Within the Type Lattice

Tensor Product Moves Diagonally in Both Directions

Taking the tensor product of a type (p1, q1) tensor with a type (p2, q2) tensor produces a type (p1 + p2, q1 + q2) tensor, moving the lattice position by (p2, q2) steps from the starting point (p1, q1), combining movement in both the p direction and the q direction simultaneously.

(p_{1}, q_{1}) + (p_{2}, q_{2}) = (p_{1} + p_{2}, q_{1} + q_{2})

Contraction Moves Diagonally Backward

Contracting a type (p, q) tensor over one upper and one lower index moves the lattice position from (p, q) to (p - 1, q - 1), a single step diagonally toward the origin, decreasing both p and q together rather than either one alone.

Raising and Lowering Moves Horizontally or Vertically

When a metric tensor is available to identify V with V*, raising an index moves the lattice position from (p, q) to (p + 1, q - 1), and lowering an index moves it from (p, q) to (p - 1, q + 1), both operations sliding diagonally along a line of constant total order p + q rather than changing the order itself.

Why Type Matters Beyond Order

Distinct Transformation Behavior at Equal Order

Two tensors sharing the same order but different type transform differently under a change of basis: a type (2, 0) tensor's components pick up the transformation matrix A twice, while a type (0, 2) tensor's components pick up the inverse matrix A^{-1} twice, and a type (1, 1) tensor's components pick up one factor of each. Order alone cannot distinguish these behaviors; only the full type specifies them.

Type Determines Valid Operations

Whether two tensors can be meaningfully contracted against one another, or whether a particular index can be raised or lowered, depends on the specific type of each tensor involved, not merely on their order, making the full (p, q) type structure the essential bookkeeping device for combining tensors correctly.

Diagrammatic Summary

The diagram shows the lattice of tensor types (p, q), with the origin (0, 0) representing scalars, the horizontal axis representing purely contravariant types, and the vertical axis representing purely covariant types, with mixed types filling the interior points.