Representation Theory: Tensor Products, Duality, Symmetric and Exterior Powers

From addition to multiplication of representations

If V and W are G-representations, the direct sum V⊕W carries the action g·(v,w)=(g·v,g·w). At the character level, direct sum corresponds to addition:

χ_{V⊕W}=χ_V+χ_W.

The tensor product V⊗W instead combines the actions multiplicatively:

g·(v⊗w)=(g·v)⊗(g·w).

At the character level,

χ_{V⊗W}(g)=χ_V(g)χ_W(g).

This simple formula is a major computational advantage. Once the irreducible character table is known, tensor-product decomposition becomes an inner-product calculation.

Why tensor products are the correct construction

A tensor product linearises bilinear behaviour. If a quantity depends linearly on v when w is fixed and linearly on w when v is fixed, the universal home for that information is V⊗W.

Representation theory needs exactly this because many symmetry-respecting interactions are bilinear. Pair two vectors, combine two state spaces, multiply two polynomial modes or couple two symmetry types: tensor products preserve the group action while linearising the combined object.

If dimV=m and dimW=n, then dim(V⊗W)=mn. That dimension multiplication is the first sanity check.

Tensor-product matrices

If ρ(g) is represented by A_g and σ(g) by B_g, then the tensor-product representation is represented by the Kronecker product

A_g ⊗ B_g.

The identities (A⊗B)(C⊗D)=AC⊗BD and tr(A⊗B)=tr(A)tr(B) explain why the tensor-product action respects the group law and why characters multiply pointwise.

Worked example: S₃ standard tensor square

The standard irreducible character of S₃ is χ=(2,0,−1) on the identity, transpositions and 3-cycles. Therefore the tensor-square character is χ²=(4,0,1).

Taking inner products with the three irreducible characters gives one copy each of the trivial, sign and standard representations. Hence

standard ⊗ standard ≅ trivial ⊕ sign ⊕ standard.

The dimensions verify the answer: 2·2=4 and 1+1+2=4.

Dual representations

Every finite-dimensional vector space V has a dual space V*=Hom(V,F), consisting of linear functionals. If V carries a G-representation, the dual must transform in the opposite direction so that evaluation remains invariant.

The dual action is defined by

(g·φ)(v)=φ(g⁻¹·v).

In matrices, if g acts on V by A_g, then it acts on V* by (A_g⁻¹)^T.

The inverse is essential. Without it, the evaluation pairing φ(v) would not be G-invariant.

The evaluation pairing

The canonical pairing V*×V→F is given by (φ,v)↦φ(v). Under the dual action,

(g·φ)(g·v)=φ(v).

This is the design constraint that determines the dual representation. Duality is therefore not an arbitrary formula. It is the unique natural way to make evaluation compatible with symmetry.

Characters of duals

For finite groups over C, representations can be made unitary. The character of the dual then satisfies

χ_{V*}(g)=overline{χ_V(g)}.

If all character values are real, V and V* have the same character and are therefore equivalent as complex representations. But equality with the dual does not automatically tell us whether the invariant bilinear form is symmetric or alternating. That distinction leads deeper into self-duality and Frobenius–Schur indicators.

Hom spaces as tensor products

There is a natural vector-space isomorphism

Hom(V,W) ≅ V*⊗W.

This becomes representation-theoretically meaningful when G acts on Hom(V,W) by conjugating the input and output actions:

(g·T)=ρ_W(g) T ρ_V(g)⁻¹.

The fixed vectors in Hom(V,W) are exactly the intertwiners Hom_G(V,W). Therefore

Hom_G(V,W) = (V*⊗W)^G.

This is a powerful unification: symmetry-respecting linear maps are invariant tensors.

Invariant bilinear forms

A bilinear form B:V×V→F corresponds to an element of V*⊗V*. The form is G-invariant when B(gv,gw)=B(v,w). Therefore invariant bilinear forms are fixed vectors inside V*⊗V*.

This is why tensor products detect self-duality. A nonzero invariant pairing V⊗V→F is the same kind of object as a nonzero intertwiner V→V*.

Symmetric and alternating tensors

The tensor square V⊗V contains two natural symmetry types under swapping the two factors.

• Symmetric tensors: unchanged when the factors are exchanged.
• Alternating tensors: change sign when the factors are exchanged.

When the characteristic is not 2, these form subspaces Sym²(V) and Λ²(V), and

V⊗V ≅ Sym²(V) ⊕ Λ²(V).

The dimensions are

dim Sym²(V)=n(n+1)/2,   dim Λ²(V)=n(n−1)/2.

Their sum is n², as required.

Why symmetric powers matter

Symmetric powers Sym^k(V) encode homogeneous polynomial behaviour of degree k. If V carries a representation, then G acts naturally on polynomial functions and symmetric tensors.

For matrix groups, symmetric powers generate families of higher-dimensional representations from a small defining representation. In the representation theory of SL₂, for example, symmetric powers of the standard two-dimensional representation provide all finite-dimensional irreducible polynomial representations.

Why exterior powers matter

Exterior powers Λ^k(V) encode alternating multilinear structure. The top exterior power Λ^n(V) is one-dimensional when dimV=n. The induced action on Λ^n(V) is multiplication by detρ(g).

This gives a representation-theoretic interpretation of determinant:

determinant is the character of the top exterior-power action.

Orientation, volume scaling and alternating geometry therefore sit naturally inside exterior representation theory.

Worked example: exterior square of a three-dimensional representation

Let V have dimension 3 with basis e₁,e₂,e₃. Then Λ²(V) has basis

• e₁∧e₂,
• e₁∧e₃,
• e₂∧e₃.

So dimΛ²(V)=3. A linear transformation A on V induces Λ²A on bivectors by

(Λ²A)(v∧w)=Av∧Aw.

If A has eigenvalues λ₁,λ₂,λ₃, then Λ²A has eigenvalues λ₁λ₂, λ₁λ₃ and λ₂λ₃. Exterior powers convert eigenvalue data multiplicatively.

Characters of symmetric and exterior squares

If χ is the character of V, then over characteristic zero the characters of the symmetric and exterior squares satisfy

χ_{Sym²V}(g) = (χ(g)² + χ(g²))/2,

χ_{Λ²V}(g) = (χ(g)² − χ(g²))/2.

Adding them gives χ(g)², the tensor-square character. These formulas are excellent decomposition tools because they let us separate symmetric and alternating components using only character values.

Worked example: S₃ standard representation again

For the two-dimensional standard representation of S₃, V⊗V decomposes as trivial⊕sign⊕standard.

Because Λ²(V) is one-dimensional, it must be either trivial or sign. Its action is determinant. Reflections/transpositions have determinant −1 and 3-cycles have determinant +1, so Λ²(V) is the sign representation.

Therefore Sym²(V) must be the remaining three-dimensional component:

Sym²(V) ≅ trivial ⊕ standard,   Λ²(V) ≅ sign.

Higher symmetric powers

Sym^k(V) consists of degree-k symmetric tensors. For a two-dimensional space, dimSym^k(V)=k+1. These spaces are central in the representation theory of SL₂ and SU₂.

If V is the defining two-dimensional representation of SL₂(C), then Sym^k(V) is irreducible of dimension k+1. This gives a clean infinite ladder of irreducibles:

2, 3, 4, 5, … dimensions via Sym^1, Sym^2, Sym^3, Sym^4, …

This is one of the simplest previews of highest-weight theory.

Higher exterior powers

Λ^k(V) vanishes for k>dimV. For 0≤k≤n, dimΛ^k(V)=C(n,k). These spaces organise oriented k-dimensional volume elements.

In geometry, differential k-forms transform through exterior powers of cotangent representations. In linear algebra, minors of a matrix are encoded by exterior-power matrices. In representation theory, fundamental representations of classical groups are often built from exterior powers.

Tensor algebra, symmetric algebra and exterior algebra

All tensor powers can be assembled into the tensor algebra

T(V)=F ⊕ V ⊕ V^{⊗2} ⊕ V^{⊗3} ⊕ ···.

Quotienting by the relations v⊗w−w⊗v gives the symmetric algebra Sym(V). Quotienting by v⊗v=0 gives the exterior algebra Λ(V).

If G acts on V, it acts compatibly on all three algebras. Representation theory can then study how each graded piece decomposes.

The symmetric group hiding inside tensor powers

On V^{⊗k}, the group G acts diagonally, while the symmetric group S_k acts by permuting tensor factors. These two actions commute.

This commuting pair is the beginning of Schur–Weyl duality. For V=C^n, the actions of GL(V) and S_k on V^{⊗k} control each other’s decomposition. Representation theory of general linear groups and representation theory of symmetric groups meet inside the same tensor space.

Symmetric and exterior powers correspond to the trivial and sign representations of S_k inside this picture. More general Young symmetrisers produce other Schur functors.

Schur functors and Young diagrams

Partitions of k index irreducible representations of S_k and also organise polynomial representations of GL_n. A Young diagram tells us how to impose mixed symmetry: symmetric in some directions, alternating in others.

Sym^k corresponds to one row of k boxes. Λ^k corresponds to one column of k boxes. General partitions interpolate between these extremes.

This is a major structural bridge. Tensor products create a huge space; symmetric-group actions classify the internal permutation symmetry; Schur functors extract canonical representation types.

Invariant tensors

A tensor fixed by the group encodes a G-invariant multilinear object. Examples include invariant inner products, volume forms, symplectic forms and structure constants.

The existence of an invariant tensor can constrain the image of the representation. For example, preserving a nondegenerate symmetric bilinear form places the image inside an orthogonal group. Preserving a nondegenerate alternating form places it inside a symplectic group.

Thus tensor invariants can reveal which classical group naturally contains the symmetry action.

Tensor products and quantum systems

Composite quantum systems use tensor products because amplitudes depend bilinearly on the component spaces before linear extension. Symmetry on the components induces symmetry on the composite space.

Representation decomposition then separates total symmetry types. In angular momentum theory, tensor products of SU₂ representations decompose according to Clebsch–Gordan rules.

The mathematics is the same structural question seen for finite groups: given irreducibles V and W, decompose V⊗W into irreducibles.

Clebsch–Gordan preview

For SU₂ irreducibles labelled by nonnegative integers n, with V_n having dimension n+1, the tensor-product rule is

V_m ⊗ V_n ≅ V_{m+n} ⊕ V_{m+n−2} ⊕ ··· ⊕ V_{|m−n|}.

For two two-dimensional defining representations V_1, this gives

V_1⊗V_1 ≅ V_2 ⊕ V_0,

that is, 2⊗2=3⊕1. The symmetric square is the three-dimensional piece; the exterior square is the one-dimensional piece.

Representation rings

Because direct sum behaves like addition and tensor product behaves like multiplication, representations can be organised into a ring-like object, the representation ring.

Its basis elements are irreducible representations. Multiplication records tensor-product decomposition. The coefficients are nonnegative integers giving multiplicities.

This converts a category of representations into an algebraic summary of how symmetry types combine.

Characters as ring homomorphism data

Characters respect both main operations:

• direct sum → addition;
• tensor product → pointwise multiplication.

Therefore the character map embeds much of the representation ring into the ring of class functions. Tensor decomposition can be studied through function arithmetic rather than matrix arithmetic.

A decomposition workflow

• Identify the characters χ_V and χ_W.
• Multiply pointwise to obtain χ_{V⊗W}.
• Take inner products with irreducible characters.
• Read off multiplicities.
• Check dimensions.
• If symmetric or exterior pieces are required, use the χ(g²) formulas or direct symmetry arguments.

Common mistakes

• Adding dimensions for tensor products: dimensions multiply.
• Forgetting the inverse in the dual action: duality must preserve evaluation.
• Assuming V⊗W is irreducible: tensor products usually decompose.
• Confusing Sym²(V) with V⊕V: symmetric square is a subquotient of V⊗V with dimension n(n+1)/2.
• Forgetting characteristic restrictions: decomposition into symmetric and alternating parts behaves differently in characteristic 2.
• Ignoring multiplicities: the same irreducible may occur more than once in a tensor product.

Practice set

1. Tensor dimension

dimV=4 and dimW=7. Find dim(V⊗W).

Answer: 28.

2. Exterior-square dimension

dimV=5. Find dimΛ²(V).

Answer: 5·4/2=10.

3. Symmetric-square dimension

dimV=5. Find dimSym²(V).

Answer: 5·6/2=15.

4. Top exterior power

If dimV=n, what representation acts on Λ^n(V)?

Answer: the one-dimensional determinant representation.

5. Invariant maps

Where do G-equivariant maps V→W appear inside V*⊗W?

Answer: as the fixed-vector subspace (V*⊗W)^G.

The deeper structural lesson

Tensor constructions show that representation theory is not only about classifying isolated actions. It is about understanding an ecosystem of actions connected by natural operations.

Duality reverses direction while preserving evaluation. Tensor products combine actions. Symmetric powers keep polynomial symmetry. Exterior powers keep alternating geometry. Hom spaces turn maps into tensors. Invariants identify symmetry-respecting structures. Representation rings record how irreducibles multiply.

The representation is not the endpoint. The real structure often appears in how representations transform, pair, tensor, split and generate one another.

Continue Representation Mathematics — Batch 02

• Induced Representations, Restriction and Frobenius Reciprocity
• Lie Group and Lie Algebra Representations | Highest Weights and Continuous Symmetry
• Modular Representation Theory | Semisimplicity, Indecomposables and Extensions
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