1.5 Tensor Representation Foundations

Tensor Representation Foundations is the body of concepts connecting tensor spaces to the theory of group representations, treating each space T^p_q(V) as carrying a natural action of the general linear group of V, and studying how that action decomposes the tensor space into smaller, irreducible pieces that cannot be broken down further. Where earlier foundational material treats a tensor as a fixed object described by components in a basis, tensor representation foundations treat the entire space of tensors of a given type as something acted upon by linear transformations of V, revealing a hidden internal structure organized by symmetry rather than by index bookkeeping alone.

The Group Action on Tensor Space

GL(V) Acting on T^p_q(V)

Every invertible linear transformation g of V, an element of the general linear group GL(V), induces a corresponding invertible linear transformation of the tensor space T^p_q(V), obtained by applying g to each of the p factors of V and the inverse transpose of g to each of the q factors of V*.

This action realizes T^p_q(V) as a representation of GL(V): an assignment of a linear transformation of the tensor space to every element of the group, compatible with composition, g_1 · (g_2 · T) = (g_1 g_2) · T.

Representation as the Structural Home of the Transformation Law

The familiar transformation law for tensor components, involving the change-of-basis matrix once for every upper index and its inverse once for every lower index, is exactly the coordinate expression of this group action; the representation-theoretic viewpoint recasts the same transformation rule as an abstract statement about how GL(V) acts on the whole tensor space at once.

Reducibility and Irreducibility

Invariant Subspaces

A subspace of T^p_q(V) is called invariant under the GL(V) action if applying any group element to any tensor in the subspace produces another tensor still within the subspace. The symmetric tensors and the antisymmetric tensors within a fixed T^p_0(V) each form such an invariant subspace, since permuting the arguments of a symmetric or antisymmetric tensor and then applying g, or applying g first and then permuting, produce the same result.

Irreducible Representations

A representation is called irreducible if it contains no invariant subspace other than the zero subspace and the whole space itself. The full tensor space T^p_0(V) is generally not irreducible for p greater than one, decomposing instead into smaller irreducible pieces, with the totally symmetric subspace and the totally antisymmetric subspace being two of the simplest such pieces.

Decomposition of Tensor Powers

Beyond Symmetric and Antisymmetric

For p greater than two, the tensor power T^p_0(V) decomposes into more than just a symmetric part and an antisymmetric part; there are additional irreducible pieces with mixed symmetry, neither fully symmetric nor fully antisymmetric, corresponding to the different ways indices can be partially symmetrized among subsets of the p slots.

Young Tableaux as an Organizing Tool

The irreducible pieces appearing in the decomposition of T^p_0(V) correspond to combinatorial diagrams called Young tableaux, which encode a specific pattern of partial symmetrization and antisymmetrization among the p indices; each distinct tableau shape of size p labels one type of irreducible piece appearing in the decomposition, and the dimension of each piece can be computed directly from the shape of its tableau.

Tensors as Carriers of Representations

Physical and Geometric Significance

Quantities that transform as tensors under a change of basis are, in representation-theoretic terms, elements of specific representations of the relevant transformation group, and identifying which irreducible representation a given physical or geometric quantity belongs to reveals structural constraints on its behavior, such as which components can be nonzero or how many independent quantities are needed to specify it.

Restriction to Subgroups

When the transformations under consideration are restricted to a subgroup of GL(V), such as the orthogonal group preserving a fixed inner product, a representation that was irreducible under the full general linear group may decompose further, or a representation that was reducible may remain so, revealing finer structure specific to the geometry preserved by that subgroup.

Relation to the Component Description

Components as a Choice of Basis for the Representation

The component array of a tensor, computed relative to a chosen basis of V, is simply the coordinate expression of the vector representing that tensor within the representation space T^p_q(V); choosing a different basis of V corresponds to acting on this vector by the corresponding element of GL(V), exactly reproducing the standard transformation law from the representation-theoretic action.

Diagrammatic Summary

The diagram shows the tensor power space T^p_0(V) decomposing under the action of GL(V) into irreducible pieces, including the totally symmetric part, the totally antisymmetric part, and, for p greater than two, additional pieces of mixed symmetry type.