Representation Theory: Leverage Map
A. EXISTENCE JUSTIFICATION
Groups describe symmetry abstractly (rotations, permutations, etc.), but we often need symmetry to act on something concrete—vectors, functions, quantum states. Representation theory exists because we needed to study symmetry through its effects on linear spaces. It was forced into existence by quantum mechanics (particles as representations of symmetry groups) and crystallography (how symmetry constrains physical structure).
The core move: Turn abstract group elements into matrices that you can compute with, while preserving the group structure.
B. CORE OBJECTS & MORPHISMS
| Object | What it is | Notation |
|---|---|---|
| Representation | A homomorphism ρ: G → GL(V), assigning each group element a matrix acting on vector space V | ρ(g) or just “V as a G-representation” |
| Carrier space | The vector space V that gets acted on | dim(V) = dimension of representation |
| Irreducible representation (irrep) | A representation with no proper invariant subspaces—can’t be broken into smaller pieces | Often labeled by indices: ρ₁, ρ₂, … or by quantum numbers |
| Character | The trace of the representation matrix: χ(g) = Tr(ρ(g)) | χ_V or χ_ρ |
| Intertwiner | A linear map T: V → W that commutes with the group action: T∘ρ_V(g) = ρ_W(g)∘T | Hom_G(V,W) = space of intertwiners |
Morphisms: Intertwiners are the structure-preserving maps. Two representations are equivalent if there’s an invertible intertwiner between them.
C. CENTRAL INVARIANTS
Character: The trace function χ(g) = Tr(ρ(g)) completely determines a representation up to equivalence. Characters are class functions—constant on conjugacy classes.
Dimension: How many degrees of freedom the symmetry acts on.
Decomposition into irreps: Every (finite-dimensional, nice) representation breaks into a direct sum of irreducibles. The multiplicities of each irrep in this decomposition are the key invariants.
What counts as “the same”: Two representations are equivalent if they’re related by a change of basis that respects the group action. Characters detect this: same character → equivalent representations.
D. SIGNATURE THEOREMS
1. Maschke’s Theorem
Every representation of a finite group (over ℂ) is completely reducible—it decomposes as a direct sum of irreducibles.
Importance: You never get “stuck” with complicated representations. Everything factors into atomic pieces. This is why representation theory is tractable.
2. Schur’s Lemma
Any intertwiner between irreducible representations is either zero (if they’re inequivalent) or a scalar multiple of identity (if they’re equivalent).
Importance: This is the rigidity theorem. It says irreps are “maximally incompatible”—you can’t partially map one into another. It’s why irreps are the natural basis for decomposition. Also: any matrix that commutes with all of a group’s action must be a scalar. This constrains what neural network layers can look like if you want equivariance.
3. Orthogonality of Characters
Characters of inequivalent irreps are orthogonal under the natural inner product on class functions.
Importance: Characters form an orthonormal basis. You can decompose any representation by taking inner products with irreducible characters—essentially Fourier analysis on groups.
E. BRIDGES TO OTHER DOMAINS
| Domain | Connection |
|---|---|
| Geometric Deep Learning | Equivariant neural networks are literally representation theory: layers must be intertwiners, features live in representation spaces |
| Quantum Mechanics | Particles are irreps of symmetry groups (Poincaré, SU(3), etc.). Spin comes from representations of SO(3)/SU(2) |
| Fourier Analysis | Fourier transform = decomposition into irreps of the translation group. Spherical harmonics = irreps of SO(3) |
| Harmonic Analysis | Generalization of Fourier to arbitrary groups |
| Algebraic Geometry | Representations of Galois groups encode deep number-theoretic information |
| Category Theory | A representation is a functor from G (viewed as a one-object category) to Vect |
| Spinors | Spin representations are projective representations of SO(n), or genuine representations of Spin(n)—the double cover |
Kat-relevant: The constraint “layer must be an intertwiner” is exactly “function must preserve structure”—same as the HoTT/protein folding framing. Representation theory tells you what maps are allowed given symmetry constraints.
F. COMMON MISCONCEPTIONS
“Representations are just matrices” — No. The representation is the homomorphism. The matrices are its image. Same representation can look very different in different bases.
“Irreducible means simple” — Irreducible representations can be high-dimensional and complex. “Irreducible” means can’t be decomposed further, not “easy.”
“All representations come from permutations” — Permutation representations are important but special. Most irreps aren’t realizable as permutations of a set.
“Characters are just a computational trick” — Characters are arguably the real invariant. The matrices are coordinate-dependent; the character is intrinsic.
Confusing representation dimension with group size — A huge group can have small irreps. The symmetric group S_n has dimension n! but its irreps range from dimension 1 to roughly √(n!).
G. NOTATION SURVIVAL KIT
| Symbol | Meaning |
|---|---|
| ρ: G → GL(V) | A representation (homomorphism from group to invertible matrices) |
| GL(V), GL(n) | General linear group: all invertible linear maps V→V (or n×n matrices) |
| χ_ρ(g) or χ_V(g) | Character: trace of ρ(g) |
| V ≅ W | V and W are isomorphic (as representations: same via intertwiner) |
| V ⊕ W | Direct sum of representations (block diagonal) |
| V ⊗ W | Tensor product of representations (dimension multiplies) |
| Res^G_H(V) | Restriction: view a G-representation as an H-representation (H ⊂ G) |
| Ind^G_H(W) | Induction: build a G-representation from an H-representation |
| V^G | Fixed points: vectors in V unchanged by all of G |
| Hom_G(V,W) | Intertwiners from V to W |
H. ONE WORKED MICRO-EXAMPLE
The group: Z/3Z = {0, 1, 2} under addition mod 3 (cyclic group of order 3)
Question: What are all irreducible representations over ℂ?
Approach: A representation assigns a matrix to each group element. Since the group is abelian, all irreps are 1-dimensional (consequence of Schur). So we need: ρ(1) = some complex number λ with λ³ = 1 (since 1+1+1 = 0 in the group).
Solution: The cube roots of unity: λ = 1, ω, ω² where ω = e^(2πi/3)
This gives three 1-dimensional irreps:
- Trivial: ρ₀(k) = 1 for all k
- ρ₁: ρ₁(k) = ωᵏ
- ρ₂: ρ₂(k) = ω²ᵏ
Verification: These are orthogonal as characters: $$\frac{1}{3}\sum_{k=0}^{2} \chi_{\rho_1}(k)\overline{\chi_{\rho_2}(k)} = \frac{1}{3}(1 + \omega\cdot\omega^{-2} + \omega^2\cdot\omega^{-4}) = \frac{1}{3}(1 + \omega^{-1} + \omega^{-2}) = 0$$
Importance: This is the discrete Fourier transform for signals with period 3. The irreps are the Fourier basis vectors. Representation theory and signal processing are the same subject.
Immediate leverage for your work:
Geometric deep learning: When a paper says “features transform in the ρ representation,” they mean features live in that carrier space. When they say “equivariant layer,” they mean intertwiner.
Spinor networks: Spinors are the spin-½ representation of the rotation group. Understanding why spinors exist (SO(3) isn’t simply connected, Spin(3) = SU(2) is its double cover) requires representation theory.
Convergence Thesis: If constraints force symmetry, representation theory tells you what structures are compatible with that symmetry. The irreps are the “legal shapes” information can take.
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