Examples are the oxygen of representation theory. The definitions are compact, but the objects they name are not. A “representation” can be a handful of matrices, a module over a group algebra, a symmetry action on solutions of an equation, or a functor into vector spaces. If you do not build examples on purpose, the subject can feel like a set of slogans.
This article is a practical recipe book: a small toolkit of constructions that reliably produces representations you can compute with, plus a worked example showing how the toolkit fits together.
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The two questions that decide which examples to build
Before choosing a construction, decide what kind of structure you want the example to highlight.
- Do you want a representation you can decompose explicitly, \to train your “splitting” instincts?
- Do you want a representation that is rigid enough to carry invariants (characters, weights, highest vectors), so you can see classification ideas in action?
You will usually get one of these cheaply and the other expensively. The recipe below lets you choose on purpose.
Recipe core: start from an action, then linearize it
The most reliable source of representations is a group action on a set or on a vector space, followed by linearization.
Permutation representations
If a group $G$ acts on a finite set $X$, you automatically get a representation on the vector space $k[X]$ with basis $\{e_x : x\in X\}$ by
What this buys you:
- Every subgroup $H\le G$ gives an action on cosets $G/H$, hence a canonical permutation representation.
- Many decompositions become combinatorial: invariant subspaces correspond \to $G$-stable partitions, constant-sum subspaces, and orbit data.
A useful habit: when you see a subgroup, mentally attach the representation $k[G/H]$. It will show up again as an induced representation in disguise.
Linear actions you already have
If $G\subset \mathrm{GL}(V)$ is given concretely, you already have a representation. The point is to add structure by turning one representation into several:
- Direct sums $V\oplus W$ and tensor products $V\otimes W$.
- Duals $V^\ast$ and Hom spaces $\mathrm{Hom}(V,W)$.
- Symmetric powers $\mathrm{Sym}^m(V)$ and exterior powers $\wedge^m(V)$.
These constructions preserve equivariance automatically and generate families of representations without new group theory.
Recipe core: induce and restrict on purpose
Restriction and induction are the two levers that let you move between “small group, explicit matrices” and “big group, structural information.”
Restriction
Given $H\le G$ and a $G$-representation $V$, you can view $V$ as an $H$-representation by forgetting part of the action. This is computationally cheap and conceptually rich, because it reveals which parts of a representation are “already visible” from a subgroup.
Typical uses:
- Reduce a decomposition problem \to a subgroup where the structure is simpler.
- Detect non-isomorphism: two $G$-representations that restrict differently \to $H$ cannot be isomorphic as $G$-representations.
Induction
Induction is the reverse direction: build a $G$-representation from an $H$-representation $W$. There are several equivalent models. A concrete one is:
- Consider functions $f: G\to W$ satisfying $f(gh)=h^{-1}\cdot f(g)$ for $h\in H$.
- Let $G$ act by left translation: $(g_0\cdot f)(g)=f(g_0^{-1}g)$.
What this buys you:
- It manufactures representations of large groups from small ones.
- It produces many irreducibles when paired with classification theorems.
- It explains permutation representations: $k[G/H]$ is the induced representation from the trivial $H$-representation.
A practical way to remember the geometry: induction spreads an $H$-module across the cosets of $H$ in $G$, with a compatibility condition.
Recipe core: work inside the group algebra when you want control
If $G$ is finite, you can treat representations as modules over $k[G]$. This point of view is especially useful when you want explicit projectors and decomposition data.
Over a field where semisimplicity holds, central idempotents in $k[G]$ carve out isotypic components. Even when semisimplicity does not hold, the group algebra still encodes:
- radicals and filtrations,
- blocks and defect,
- extension data through module homomorphisms.
The recipe is:
- Put the representation into the algebraic language.
- Use the algebra to compute invariants and maps.
This reduces many “matrix” questions \to “ideal” questions.
Worked example: build and decompose the standard representation of $S_3$
The symmetric group $S_3$ is the smallest non-abelian group, which makes it a perfect laboratory. Work over a field $k$ of characteristic not dividing $6$, so the clean decomposition theorems apply.
Step: start from an action
$S_3$ acts on the set $\{1,2,3\}$ by permuting labels. Linearize this action to get the permutation representation on $k^3$ with basis $e_1,e_2,e_3$.
This representation is already informative, but it has an obvious invariant direction: the “all ones” vector
It is fixed by every permutation, so $ku$ is a copy of the trivial representation.
Step: remove the trivial piece to expose a new one
Consider the subspace
This is the kernel of the “sum” map $k^3\to k$, hence $S_3$-stable because permutations preserve the sum. We have a direct sum decomposition
The representation on $W$ is the two-dimensional standard representation of $S_3$. You built it without guessing matrices: it emerged naturally from the permutation action.
Step: compute enough to recognize irreducibility
To see that $W$ is irreducible over $k$ in this characteristic regime, there are several quick tests.
One test uses characters, but you do not need full tables. Observe:
- A transposition in $S_3$ fixes one basis vector and swaps the other two. In $k^3$ it has trace $1$. On the trivial line $ku$, it has trace $1$. So on $W$, it has trace $0$.
- A 3-cycle has trace $0$ on $k^3$ because it permutes basis vectors in a 3-cycle. On $ku$, it has trace $1$. So on $W$, it has trace $-1$.
Those traces distinguish $W$ from both the trivial and the sign representation, so $W$ cannot have a one-dimensional invariant subspace unless it splits as a sum of those one-dimensional representations. But trace data already conflicts with that possibility. In this setting, the clean conclusion is that $W$ is irreducible.
Step: recover the sign representation from the same action
The determinant map on permutation matrices gives a one-dimensional representation: $\mathrm{sgn}: S_3\to \{\pm 1\}\subset k^\times$. This is a different example built from the same object $S_3$, and it highlights a general method:
- if your group acts on a vector space, look for multiplicative invariants of the action such as determinant, orientation, or volume scaling.
Now you have the full irreducible list for $S_3$ in this regime: trivial, sign, and the standard two-dimensional representation.
The example demonstrates the recipe in miniature:
- start from an action,
- linearize,
- identify an invariant subspace,
- split off what you understand,
- iterate.
Beyond finite groups: the same recipes reappear
The core constructions above are not confined to finite groups.
Lie algebras and highest vectors
For Lie algebras, the most productive recipe is often:
- choose a Cartan subalgebra and a decomposition into raising and lowering parts,
- build a module from a chosen highest vector,
- generate the rest by applying lowering operators,
- read structure from weight spaces.
Even without the full classification theorems, this recipe produces explicit representations you can compute with and reason about.
Compact groups and inner products
For compact groups, you can average an inner product to make the action unitary. That creates strong structure:
- orthogonal complements of invariant subspaces are invariant,
- decompositions behave like Fourier analysis on the group.
The recipe is the same: “find an action, then average the structure you need,” but the averaging is an inner product rather than a projection.
A checklist you can reuse whenever you need an example quickly
When you need an example in representation theory, pick a line from the checklist and follow it.
- Use a group action on a set $X$, then linearize \to $k[X]$.
- Use a subgroup $H$ and build $k[G/H]$ as a standard induced representation.
- Take a known representation and generate more using $V^\ast$, $V\otimes W$, $\wedge^m V$, $\mathrm{Sym}^m V$, and $\mathrm{Hom}(V,W)$.
- Translate the problem into the group algebra $k[G]$ when you want idempotents, ideals, and endomorphism rings to do the work.
- When the setting supports it, use traces on a few conjugacy classes to identify components without full tables.
If you practice these recipes, representation theory becomes a subject where you can manufacture the objects you need rather than waiting for them to appear.
Closing: why example-building is not optional here
In many areas of mathematics, definitions already carry enough intuition that examples are confirmations. In representation theory, examples are often the definition in action. They are where you learn what the invariants actually see, where decompositions actually occur, and where subtle failures show up when hypotheses shift.
The recipes in this article are not a replacement for deeper structure theorems. They are the scaffolding that lets you understand those theorems as answers to questions you can already ask concretely.
Books by Drew Higgins
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