advanced29 min studyLesson 511 in Mathematics

Linear representations and intertwiners

Turn group multiplication into compatible linear maps and identify the maps that preserve an action.

Before you begin

On this page

The question for today

What you’ll understand

Check a representation from generators and relations.
Solve intertwining equations and distinguish equivalence from equal matrices.
Find invariant information under a simultaneous change of basis.

A group describes how symmetries compose. A linear representation makes those compositions act on a vector space, so that matrix methods can expose patterns hidden in a multiplication table. Throughout this module groups are finite and representations are finite dimensional; the field is complex unless another field is explicitly named. The field matters: a real action can remain inseparable into invariant lines even when its complexification splits. We begin with the actual compatibility equations before using traces to simplify their solutions.

The multiplication law must survive

A representation consists of a vector space and a homomorphism from the group to its invertible linear maps. The identity must act as the identity operator, and a product must act by the corresponding product of operators in the declared order. In particular, assigning an arbitrary invertible matrix to every group element does not define a representation. The assignments must satisfy all relations. For a cyclic group of order three, choosing a matrix whose cube is the identity suffices because every element is a power of the chosen generator.

For a presented group, check each defining relation on the proposed generator matrices. The triangle symmetry group has a rotation of order three and a reflection of order two, with conjugation by the reflection reversing the rotation. A planar rotation through 120 degrees and reflection across the horizontal axis satisfy these relations over both the real and complex numbers. Faithfulness is a separate question: the trivial representation satisfies every relation while sending all elements to the same operator. A representation need not preserve distinctions between group elements.

\rho:G\longrightarrow\operatorname{GL}(V),\qquad \rho(gh)=\rho(g)\rho(h),\quad \rho(e)=I,\quad \rho(g^{-1})=\rho(g)^{-1}

These equations fix a left action on column vectors. The rightmost operator acts first.↔ Scroll wide equations horizontally.

Intertwiners preserve an entire action

An intertwiner from one representation to another is a linear map that commutes with the actions in the appropriate spaces. It can be rectangular and need not be invertible. Its kernel and image are invariant subspaces, because applying a group element before or after the map gives the same result. Thus an intertwiner carries structural information even when it loses dimensions. An invertible intertwiner is an equivalence of representations; its existence says that the actions differ only in coordinates.

For representations specified by generators, intertwining the generators automatically intertwines their products and inverses. This reduces an apparently large collection of constraints to linear equations in the entries of the unknown map. Solving those equations is often easier than guessing a basis. When the two representations coincide, the intertwiners form an algebra called the commutant. Its dimension and multiplication reveal repeated components, although conclusions about irreducibility need the field and decomposition hypotheses developed in the next lessons.

T\rho(g)=\sigma(g)T\quad(g\in G),\qquad T\text{ invertible}\Longrightarrow \sigma(g)=T\rho(g)T^{-1}

An intertwiner compares two actions. A single simultaneous conjugation, rather than a different basis change for each group element, expresses equivalence.↔ Scroll wide equations horizontally.

Coordinates change while structure survives

Trace, determinant, dimension, and the dimensions of invariant subspaces survive a simultaneous basis change. Individual entries do not. If a representation is a direct sum, its generator matrices become block diagonal in a suitable common basis, and every group element respects the same blocks. Diagonalizing just one matrix is insufficient for a nonabelian group, since another generator can exchange its eigenspaces. The goal is a common invariant decomposition, not a collection of unrelated diagonal forms.

Permutation actions supply representations without initially choosing numerical matrices: take one basis vector for every point and permute that basis. The sum of all basis vectors is fixed, while differences often describe changing configurations. This construction connects combinatorial actions to linear algebra and later turns fixed-point counts into character values. Keeping the action convention explicit prevents accidental inverses when passing between a permutation of points and the induced action on functions, where pullback usually evaluates a function at the inverse-transformed point.

Work it through

Solve the commutant of a coordinate swap

Let the nonidentity element of $C_2$ act on $\mathbb C^2$ by $S=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ . Find every endomorphism intertwining this action with itself.

Write $T=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ . Then $TS=\begin{pmatrix}b&a\\d&c\end{pmatrix}$ whereas $ST=\begin{pmatrix}c&d\\a&b\end{pmatrix}$ .
Equality gives $c=b$ and $d=a$ , leaving two independent complex parameters. Thus every intertwiner has the form $aI+bS$ .
On the vectors $(1,1)$ and $(1,-1)$ , this operator has eigenvalues $a+b$ and $a-b$ . It is an equivalence precisely when neither vanishes; otherwise it is still an intertwiner, possibly a projection or the zero map.

The result

The commutant is two dimensional. The swap action has a common fixed line and an antisymmetric line, and its intertwiners scale those lines independently.

The living notebook

Explore the model

Interactive lab

Interactive lab · vectors

A cyclic generator rotates a vector

Pause and Reset, then set both angle $120$ degrees and magnitude $1$ . Predict projections $(-1/2,\sqrt3/2)$ and compare the readouts. Reset and set both angle $-120$ degrees and magnitude $1$ for the inverse rotation. The instrument shows a single vector, not repeated matrix multiplication; compute three successive 120-degree rotations on paper to check the order-three relation. Control changes alone retain elapsed time.

Vectorx / y projectionsTrace marker

x component: 1.393
y component: 0.975
Magnitude: 1.7

Angle35°Magnitude1.7

t = 0 s

Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe angle of one image becomes a representation only after the generator relation is checked.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Equivalence requires one common invertible intertwiner. Matching dimensions or generator determinants is too weak, and a nonzero linear map may have a kernel. The common invertible intertwiner is the definition of equivalence.

Your turn

Try it yourself

Let $C_3$ act on $\mathbb C^2$ with generator $D=\operatorname{diag}(\omega,\omega^2)$ , where $\omega=e^{2\pi i/3}$ . Determine all maps commuting with $D$ and decide whether swapping the two coordinates is such a map.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Compare the two off-diagonal entries of TD and DT, using the fact that the two eigenvalues are distinct.

Compare with the solution

For $T=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ , commutation forces $b(\omega^2-\omega)=0$ and $c(\omega-\omega^2)=0$ . Hence $b=c=0$ , while $a,d$ are arbitrary. The coordinate swap is not in this commutant. It instead intertwines the displayed action with the action whose generator has the two diagonal entries reversed; these are equivalent actions in differently ordered bases.

Keep the idea

Take it with you

A representation respects products, identities, and inverses.
Intertwiner equations are linear even though equivalence asks for invertibility.
Equivalent representations use one simultaneous basis change.

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