§26

A glimpse of representation theory

To "linearize" a finite group G — find a faithful homomorphism $ρ : G \to G L_{n} (C)$ — is the entrance to representation theory. Translating group questions into linear algebra is one of the great inventions of late-19th-century mathematics (Frobenius, Schur).

Definition 26.1 — representation

A representation of G is a homomorphism

ρ : G \to G L_{n} (C)

sending g to an invertible complex matrix. n is the dimension. ρ is irreducible if no non-trivial subspace is invariant under all ρ(g). Every representation of a finite group is a direct sum of irreducibles (Maschke's theorem).

26.1 Characters

The character $χ_{ρ} (g) = tr ρ (g)$ is a class function on G (depends only on conjugacy class). Irreducible characters form an orthonormal basis: $⟨ χ_{ρ}, χ_{ρ^{'}} ⟩ = \frac{1}{∣ G ∣} g \in G \sum χ_{ρ} (g) \overline{χ_{ρ^{'}} (g)} = δ_{ρ ρ^{'}} .$ This is "Fourier analysis on G", perfectly analogous to Fourier series on $R / Z$ .

26.2 1-dimensional representations of G

χ	1	R	R²	other conj. classes...
χ_triv	1	1	1	1 ...
χ_sgn	1	−1	1	±1 ...
χ_3, χ_4, ...	d	tr(ρ(R))	tr(ρ(R²))	...

G has exactly two 1-dimensional irreducible representations: the trivial rep and the sign rep (mapping odd permutations to −1). This matches G^ab = ℤ/2. All other irreducibles are higher-dimensional (≥ 2), reflecting how strongly non-Abelian G is.

Any 1-dim irreducible $ρ : G \to C^{*}$ factors through $G^{ab}$ since $C^{*}$ is Abelian. So the number of 1-dim irreducibles = |G^ab|. For the cube, G^ab = ℤ/2, giving exactly two.

26.3 Maschke + Schur — the two pillars

Maschke's theorem (1899)

Every (finite-dim) complex representation of a finite group G is completely reducible: it decomposes as a direct sum of irreducibles. Key step: given a G-invariant subspace W ⊆ V, construct an invariant complement via the averaging projector

P = \frac{1}{∣ G ∣} \sum_{g \in G} ρ (g) P_{0} ρ (g)^{- 1}

Schur's lemma (1905)

Let

ρ_{1}, ρ_{2}

be irreducible representations of G and

T : V_{1} \to V_{2}

a G-equivariant linear map (

T ρ_{1} (g) = ρ_{2} (g) T

). Then:
(a) if

ρ_{1} \neq ≅ ρ_{2}

, then

T = 0

;
(b) if

ρ_{1} = ρ_{2}

, then

T = λ I

(a scalar).

These two deceptively simple statements imply nearly every key result in finite-group representation theory:

ρ \in G \sum d_{ρ}^{2} = ∣ G ∣, ∣ G ∣ = # {conj. classes}

Left: sum of squares of irreducible dimensions equals |G|. Right: the number of irreducibles equals the number of conjugacy classes. For the cube:

ρ \sum d_{ρ}^{2} = 43, 252, 003, 274, 489, 856, 000 with ∣ G ∣ \approx 81, 120

This puts a non-trivial constraint on 81,120 non-negative integers $d_{1}^{2}, d_{2}^{2}, \dots$ summing to 4.3 × 10¹⁹. The abelianization G^ab = ℤ/2 forces two of them to be 1; the remaining 81,118 are all ≥ 2, with average dimension $∣ G ∣/∣ G ∣ \approx 23, 000$ .

26.4 Orthogonality relations

Irreducible characters form an orthonormal basis of the space of class functions on G. The two dual orthogonality relations:

g \in G \sum χ_{ρ} (g) \overline{χ_{ρ^{'}} (g)} = ∣ G ∣ \cdot δ_{ρ ρ^{'}} (first orthogonality)

ρ \in G \sum χ_{ρ} (g) \overline{χ_{ρ} (h)} = {∣ C_{G} (g) ∣ 0 g \sim h g \neq \sim h (second orthogonality)

The first says irreducible characters are orthonormal. The second is "column orthogonality", giving $∣ C_{G} (g) ∣ = ∣ G ∣/∣ class (g) ∣$ (centralizer × class = |G|, §8.1). Together, the character table is a strong group invariant, equivalent to G for Abelian groups and almost equivalent in the non-Abelian case.

26.5 Fourier analysis & random walks

The §24 random-walk bound comes from representation theory. Given a probability measure μ on G (the one-step distribution), its group Fourier transform is

\overset{μ}{^} (ρ) = g \in G \sum μ (g) ρ (g) \in M_{d_{ρ}} (C) .

"t-step distribution" = t-fold convolution of μ, and convolution becomes multiplication under Fourier: $μ^{t} (ρ) = \overset{μ}{^} (ρ)^{t}$ . Applying Parseval:

d_{T V}^{2} (μ^{t}, Unif) \leq \frac{1}{4} ρ \neq = triv \sum d_{ρ}^{2} \overset{μ}{^} (ρ)^{t}_{op}^{2} .

Each $\overset{μ}{^} (ρ)$ has operator norm ≤ 1, with equality only at ρ = trivial. So on non-trivial ρ this is a strict contraction whose rate equals the largest second eigenvalue. This is the group-theoretic heart of Diaconis's 1980s shuffle revolution: mixing rate = worst operator norm over non-trivial irreducibles.

26.6 G's character table — an open object

The complete classification of G's irreducibles (full list of ρ's plus character values) has never been written out. Known facts:

Number of irreducibles = number of conjugacy classes $\approx 81, 120$ (Frobenius identity)
Exactly 2 one-dim irreducibles (trivial + sign), since $G^{ab} = Z /2$
Largest irreducible dimension is unknown (estimated $\sim 1 0^{5}$ ), arising from non-trivial induced representations of $A_{8} \times A_{12}$
All major CAS (GAP, Magma, SageMath) time out / run out of memory on the full character table

"G's character table is a perfect computational challenge: the group structure is known, the algorithms are known (Burnside, Brauer, Dixon-Schneider), but the computation exceeds present-day resources."

— Joyner, on computational limits

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