16.21.5 Tensor Alternating Tensor Representation Role

Tensor Alternating Tensor Representation Role is the function that alternating tensors and exterior powers serve within representation theory, providing a natural family of representations of the general linear group and the symmetric group built directly from a vector space's exterior powers, with the alternating property itself corresponding to one of the simplest and most fundamental irreducible representations of the symmetric group. It situates alternating tensors within the broader classification of representations arising from tensor powers of a vector space.

Exterior Powers as Representations of the General Linear Group

The Natural Action on Λᵏ(V)

If V carries an action of the general linear group GL(V) by linear automorphisms, this action extends naturally to each exterior power Λᵏ(V) via the induced exterior power map, since any invertible linear map T on V induces a corresponding invertible map Λᵏ(T) on Λᵏ(V):

This makes Λᵏ(V) a representation space of GL(V), often denoted using the same exterior power notation to emphasize its dual role as both an algebraic construction and a representation.

Irreducibility of Exterior Power Representations

For the standard representation of GL(V) on V itself, the exterior power representations Λᵏ(V) are irreducible for every k from 0 to n, meaning they contain no proper nonzero invariant subspaces under the group action. This irreducibility is a direct structural consequence of the alternating property: the strong constraints imposed by antisymmetry leave no room for a smaller invariant piece to split off within a fixed exterior power degree.

Alternating Tensors as a Symmetric Group Representation

The Sign Representation

Considering instead the action of the symmetric group Sₖ permuting the k factors of a tensor product V ⊗ ... ⊗ V, the subspace of alternating tensors corresponds to the isotypic component associated with the sign representation of Sₖ, the one-dimensional representation in which each permutation σ acts by multiplication by sgn(σ).

Contrast With the Trivial Representation

Where symmetric tensors correspond to the trivial representation of Sₖ, in which every permutation acts as the identity, alternating tensors correspond instead to the sign representation, providing a direct representation-theoretic explanation for why symmetric and alternating tensors behave as complementary, oppositely signed structures under index permutation.

Connection to Schur-Weyl Duality

Decomposing Tensor Powers

Schur-Weyl duality describes how the k-fold tensor power V^⊗k decomposes simultaneously under the commuting actions of GL(V) and the symmetric group Sₖ, with each irreducible GL(V) representation appearing paired with a corresponding irreducible Sₖ representation, both indexed by the same partition of k. Within this decomposition, the exterior power Λᵏ(V) corresponds to the partition consisting of a single column of length k, paired precisely with the sign representation of Sₖ.

Symmetric Powers as the Dual Extreme

The symmetric power Symᵏ(V) occupies the opposite extreme of this classification, corresponding to the partition consisting of a single row of length k, paired with the trivial representation of Sₖ. Alternating tensors and symmetric tensors thus sit at the two structural extremes of the Schur-Weyl decomposition of tensor powers, with more general irreducible pieces, corresponding to intermediate partition shapes, filling the space between them.

Applications in Representation-Theoretic Contexts

Highest Weight Theory

In the study of highest weight representations of the general linear or special linear group, the exterior powers Λᵏ(V) provide some of the simplest and most explicitly describable fundamental representations, serving as building blocks from which more general irreducible representations can be constructed using Schur functors associated with more elaborate partition shapes.

Plethysm and Character Computations

The characters of exterior power representations are computed using elementary symmetric polynomials, providing a direct combinatorial link between the representation-theoretic role of alternating tensors and classical symmetric function theory, since the character of Λᵏ(V) as a representation of GL(V) is precisely the k-th elementary symmetric polynomial evaluated at the eigenvalues of the acting group element.

Significance of the Role

The representation-theoretic role of alternating tensors situates exterior powers as irreducible representations of the general linear group and as the sign-representation isotypic component under symmetric group permutation, occupying one structural extreme of the broader Schur-Weyl decomposition of tensor powers. This role connects the antisymmetry properties studied throughout alternating tensor theory to the classification of irreducible representations, symmetric function theory, and the combinatorics of partitions used throughout representation theory.