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Tensor representation

A representation of a general linear or matrix group obtained from finite tensor products of a fundamental representation and its dual, including their irreducible factors.

Version
v2 · 2026-09-06 · History
Domain-specific #
2944
Origin domain
mathematics
Subdomain
representations of general linear and classical groups
Aliases
Mixed tensor representation, Rational tensor representation

Core Idea

Tensor representation is a representation of a general linear or matrix group obtained from finite tensor products of a fundamental representation and its dual, including their irreducible factors.

Starting from the defining representation V of GL(V) and its dual V, finite tensor powers V^⊗p ⊗ (V)^⊗q inherit diagonal group actions. Subrepresentations, quotients, direct sums, and irreducible factors generated from these tensors form the tensor or rational representation family. Schur functors and Young diagrams organize polynomial irreducibles; mixed tensors incorporate dual factors.

Scope of Application

The abstraction recurs literally within general linear and classical matrix groups whose representations are generated from defining modules and their duals. The following habitats preserve the same recognition machinery; they are not invitations to extend the name metaphorically.

  • Polynomial GL representations. Schur functors extract irreducibles from V tensor powers.
  • Rational representations. mixed primal and dual powers allow negative weight components.
  • Orthogonal groups. trace-free symmetric tensors give standard tensor representations.
  • Symplectic groups. contractions and symmetry types organize constituents.
  • Invariant theory. tensor contractions produce equivariants and invariants.

Clarity

The word tensor may refer to a tensor space, a tensor field, or this representation family. Membership is categorical: the action must be obtained from the defining action and dual through finite tensor operations and the allowed subquotients. The acting group and base field must be stated.

Manages Complexity

Tensor generation replaces an open-ended search over representations with combinatorics of degrees, symmetries, contractions, and highest weights. It also makes transformation behavior explicit and computable.

The compression remains accountable because each simplification has a named failure condition. Disagreement can be localized to a missing role, an invalid assumption, an ambiguous measurement, or a neighboring abstraction instead of being hidden inside an unanalyzed label.

Abstract Reasoning

R1. Fix the group and its defining module. R2. Construct the diagonal action on specified primal and dual tensor powers. R3. Identify invariant subspaces or quotient modules. R4. Apply Schur or contraction operators with characteristic hypotheses attached. R5. Distinguish tensor, spin, and other representation families before claiming completeness.

Knowledge Transfer

The construction transfers literally among matrix groups and fields supporting the same tensor category. Representation and symmetry are broader parents; using tensor notation in a neural model or physical equation does not automatically define a tensor representation of a group.

The transfer boundary is explicit: DOMAIN-SPECIFIC PASS / PRIME FAIL: The construction recurs across tensor degrees, dual factors, Schur functors, Young tableaux, and suitable matrix groups. Literal recognition retains the specialist vocabulary and validity conditions of representation theory; outside that setting only broader parent operations transfer.

Relationships to Other Abstractions

Local relationship map for Tensor representationParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Tensor representationDOMAINPrime abstraction: Symmetry — presupposesSymmetryPRIMEPrime abstraction: Representation — is a kind ofRepresentationPRIME

Current abstraction Tensor representation Domain-specific

Parents (2) — more general patterns this builds on

  • Tensor representation is a kind of Representation Prime

    Representation (prime:representation).

  • Tensor representation presupposes Symmetry Prime

    Symmetry (prime:symmetry).

Hierarchy paths (2) — routes to 2 parentless roots

Neighborhood in Abstraction Space

Tensor representation sits in a sparse region of the domain-specific corpus (70th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08