Representation theory rewards precision, but it also punishes casual habits more sharply than many neighboring subjects. The reason is structural: the objects live at the intersection of algebra, geometry, and linear algebra, and the meaning of a statement can change when you move even slightly across that intersection.
This article collects common mistakes that appear in self-study, in seminar notes, and in early research work, and it shows how to avoid them with simple checks. The goal is not to make you cautious in a timid way. The goal is to make you fast in a disciplined way.
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Confusing “irreducible” with “indecomposable”
This is the most frequent conceptual slip because in semisimple settings the words collapse into one idea. Outside semisimple settings, they separate sharply.
- Irreducible means the representation has no nontrivial invariant subspace.
- Indecomposable means the representation is not a direct sum of two nonzero invariant subspaces.
A representation can be indecomposable without being irreducible: it can contain invariant subspaces but still fail to split.
How to avoid the mistake:
- Before using a decomposition argument, check whether your category is semisimple.
- For finite groups, check whether $\mathrm{char}(k)$ divides $|G|$.
- For algebras, check whether the algebra has nilpotent ideals or a nontrivial Jacobson radical.
If semisimplicity is not in place, treat “composition series” and “direct sum decomposition” as different operations.
Using Maschke’s theorem without checking divisibility
Many standard moves in finite group representation theory are disguised uses of averaging. Averaging usually includes a factor of $1/|G|$. If that denominator does not exist in your field, the move is invalid.
Typical symptom:
- You write “take a complement” or “take an invariant projection” as if it were automatic.
Fast check:
- Write down exactly where you divided by $|G|$, even if it was only implicit in “average over the group.”
- If the field does not support that division, switch your plan: use filtrations, radicals, blocks, or cohomological obstructions instead of complements.
Assuming characters classify representations in every field
Characters are powerful, but their power is not unconditional.
Over $\mathbb{C}$ for finite groups, the character determines the representation up to isomorphism. In other settings, trace data can be too weak because it cannot see nilpotent structure.
How to avoid the mistake:
- Know the regime where “character determines representation” is true.
- If you are not in that regime, treat characters as invariants, not classifiers.
A practical heuristic:
- If your matrices can be put into Jordan form with nontrivial nilpotent part, then trace-based invariants cannot detect that nilpotent part.
Forgetting that “same matrices” is not the definition of isomorphism
Two representations $\rho,
ho': G\to \mathrm{GL}(V)$ are isomorphic if there exists an invertible linear map $T: V\to V$ such that
for all $g\in G$.
The common error is to compare matrices in a fixed basis and conclude “different matrices, different representation” or “same matrices, same representation” without checking how the basis choice is controlling the picture.
How to avoid the mistake:
- Translate “isomorphic” into “conjugate as homomorphisms,” meaning $\rho'(g) = T
\rho(g)T^{-1}$.
- When you compute, decide whether you are working up to conjugacy or in a fixed basis.
This is not pedantry. Many classification results are statements about conjugacy classes of homomorphisms, not about raw matrices.
Misapplying Schur’s lemma when the field is not algebraically closed
A common slogan: “Endomorphisms of an irreducible representation are scalars.” That is true over an algebraically closed field in the finite-dimensional setting, but it changes if the field is not algebraically closed.
Over a general field $k$, $\mathrm{End}_G(V)$ for an irreducible $V$ is a division algebra over $k$, which can be larger than $k$.
How to avoid the mistake:
- If the field is not algebraically closed, treat Schur’s lemma as “division algebra,” not “scalars.”
- If you want “scalars,” add the hypothesis or base change to an algebraic closure and track what changes.
This matters whenever you are computing commutants, multiplicities, or decompositions with symmetry.
Mixing left and right actions when using group algebras
When you rewrite representations as modules over $k[G]$, you must choose whether you are using left modules or right modules. Many formulas change by inverses when you switch, and it is easy to slide between them without noticing.
Typical symptom:
- You define the module action by $g\cdot v =
\rho(g)v$ and later write formulas that treat $v\cdot g$ as if it were the same.
How to avoid the mistake:
- Decide once: left module convention is standard in many texts.
- When you define induced representations as function spaces, write the equivariance condition explicitly and check where inverses appear.
- If a sign or inverse appears “mysteriously,” the cause is often a \left/right mismatch.
Confusing restriction with taking invariants
Restriction is a change of group, not a change of vector space. Invariants are a change of vector space, not a change of group. People sometimes blur them because both operations “simplify.”
- Restriction: view $V$ as an $H$-representation for $H\le G$, keeping the same $V$.
- Invariants: take $V^H = \{v\in V : h\cdot v = v \text{ for all } h\in H\}$, shrinking the space.
How to avoid the mistake:
- When you write $V\downarrow_H$, say in words: “same vectors, smaller group.”
- When you write $V^H$, say in words: “smaller vectors, same action.”
These are different functors with different exactness properties. Confusing them can derail a proof silently.
Treating “tensor product” as merely multiplying dimensions
Tensor products in representation theory are not just size changes; they change symmetry content. The biggest mistake is to treat $V\otimes W$ as a black box and assume it behaves like a direct sum.
How to avoid the mistake:
- When you form $V\otimes W$, immediately ask what the action is: $g\cdot (v\otimes w) = (g\cdot v)\otimes (g\cdot w)$.
- Check whether the tensor product introduces invariants: $G$-fixed vectors in $V\otimes W$ correspond \to $G$-equivariant maps $V^\ast\to W$.
- Use weight decompositions or character multiplication rules when those tools apply.
A small, reusable insight: invariants in $V\otimes V^\ast$ always contain the identity map, which is why endomorphism rings keep appearing.
Misusing orthogonality relations as if they were purely formal identities
Character orthogonality is not a symbolic trick you can apply anywhere. It depends on a very specific setup: finite group, class functions, and an inner product that is built from averaging over the group.
Typical mistakes include:
- Using orthogonality while working over a field where averaging is invalid.
- Treating “character inner product” as meaningful without checking that characters are defined as class functions into a field where traces behave as expected.
- Forgetting that the orthogonality relations live in the space of class functions, so they do not automatically control module extensions or nilpotent structure.
How to avoid the mistake:
- State the field and the averaging inner product before invoking orthogonality.
- Use orthogonality to detect multiplicities inside semisimple decompositions, not to prove semisimplicity itself.
Misreading “semisimple” as “diagonalizable”
Semisimplicity is a statement about splitting of modules, not a statement that every operator diagonalizes.
A representation can be semisimple even when particular group elements act by matrices that are not diagonalizable over the base field. The correct distinction is:
- Semisimple category: every short exact sequence splits.
- Diagonalizability: a property of a single linear operator over a chosen field.
How to avoid the mistake:
- Keep “module-theoretic” language for semisimplicity: splitting, direct sums, complements.
- Keep “operator-theoretic” language for diagonalization: eigenvalues, Jordan blocks, minimal polynomials.
Mixing the languages leads to incorrect inferences about decompositions.
Ignoring topology when dealing with Lie groups or compact groups
When representations involve Lie groups, continuity is part of the data. A purely algebraic homomorphism into $\mathrm{GL}(V)$ might exist, but it might not be continuous, and many theorems assume continuity.
How to avoid the mistake:
- State whether your representation is continuous, smooth, or analytic when the group is a topological group.
- When using decomposition results for compact groups, check that the inner product averaging step is allowed, which requires compactness and continuity.
A sign that topology matters: when the proof uses integration over the group, the representation must interact well with that integration.
A compact “sanity checklist” before you commit \to a proof
When you are about to use a standard theorem or a standard computation, run the checklist. It catches most mistakes quickly.
- Field check: characteristic and algebraic closure status.
- Semisimplicity check: what theorem guarantees splitting in your setting?
- Category check: are you in group representations, Lie algebra modules, or modules over a general algebra?
- Functor check: are you restricting, inducing, taking invariants, or taking coinvariants, and do you know which ones are exact?
- Matrix check: are you working up to change of basis, or in a fixed basis?
This checklist is short because it is not a substitute for understanding. It is a guardrail that keeps understanding from being wasted.
Closing: precision is not a burden here, it is speed
Representation theory becomes genuinely enjoyable once the “dangerous shortcuts” are replaced by “safe shortcuts.” The safe shortcuts are not memorized tricks. They are small checks that prevent category-level errors: using the wrong field hypothesis, confusing splitting with diagonalization, forgetting topology, or swapping invariants with restriction.
If you build the habit of stating your regime and running the sanity checklist, you will find that proofs become faster, computations become more reliable, and you will be free to focus on the real question: what symmetry is doing, and what information the representation is designed to carry.
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