REFERENCE SHEET

Direct sums, tensor products & duals

Build new representations and track how their dimensions and characters change.

Before this Representations & invariant subspaces, Characters & decomposition

χV⊕W = χV + χW; χV⊗W = χV χW

Definitions

Direct sum

Act independently on V⊕W: g·(v,w)=(g·v,g·w). Dimensions add.

Tensor product

The vector space generated by symbols v⊗w subject to bilinearity. The action is g·(v⊗w)=(g·v)⊗(g·w); finite dimensions multiply.

Dual representation

On V*=Hom(V,F), define (g·ℓ)(v)=ℓ(g⁻¹·v). The inverse makes this a left action.

Key theorems

Character identities

For finite-dimensional representations, direct sums add traces and tensor products multiply them.

Dual characters

For a finite group over ℂ, χV*(g)=χV(g⁻¹), which is the complex conjugate of χV(g).

Invariant tensors

The invariant subspace of V*⊗W identifies with HomG(V,W), the space of intertwiners.

Notable applications

Coupled systems

Tensor products describe combined state spaces on which one symmetry acts simultaneously.

Invariant quantities

Trivial summands in tensor products reveal symmetry-invariant combinations.

Work it out

Try before revealing

01 / Compute

For the sign representation ε of C₂, identify ε⊗ε.

Hint

Square the character values.

Solution

Its character is (1,−1)²=(1,1), so it is the trivial representation.

02 / Compute

If dim V=2 and dim W=3, what are dim(V⊕W) and dim(V⊗W)?