Questions: Representations of Cyclic Groups

Question 1

How many non-isomorphic irreducible complex representations does ℤ/6ℤ have, and what are their dimensions?

Model answer: Exactly 6 irreducible representations, all of dimension 1.

A cyclic group of order n has n conjugacy classes (each element is its own class, since the group is abelian). The number of irreducible representations equals the number of conjugacy classes, so there are n = 6 irreducibles. Since the group is abelian, Schur's lemma forces all irreducibles to be 1-dimensional. The sum-of-squares check confirms: 1² × 6 = 6 = |G|.

Question 2

The representation ρ_k: ℤ/nℤ → GL₁(ℂ) defined by ρ_k(g) = e^{2πik/n} for a generator g is irreducible for all k = 0, …, n−1.

• T. True
• F. False

Answer: True

Each ρ_k is a group homomorphism from ℤ/nℤ to ℂ* (since (e^{2πik/n})ⁿ = 1). These are 1-dimensional, hence automatically irreducible (no proper nonzero subspaces exist). They are pairwise non-isomorphic because ρ_k(g) ≠ ρ_ℓ(g) when k ≠ ℓ (mod n). Since there are exactly n of them and n = |G| = Σ dᵢ² requires n summands of 1², this is the complete list.

Question 3

What is the character table of ℤ/3ℤ?

• A. A 3×3 identity matrix
• B. A 3×3 matrix with all entries equal to 1
• C. The 3×3 DFT matrix with entries ω^{jk} where ω = e^{2πi/3}
• D. A 3×3 matrix with entries ±1

Answer: (C)

The character table has rows indexed by irreducible representations ρ₀, ρ₁, ρ₂ and columns by group elements e, g, g². The entry χ_k(gʲ) = ω^{jk} where ω = e^{2πi/3}. This gives the matrix [[1,1,1],[1,ω,ω²],[1,ω²,ω]], which is precisely the 3×3 discrete Fourier transform matrix. The orthogonality relations for characters correspond to the unitarity of the DFT matrix (after scaling by 1/√3).

Question 4

Over ℝ (rather than ℂ), the cyclic group ℤ/4ℤ has an irreducible representation of dimension 2.

• T. True
• F. False

Answer: True

Over ℂ, the four irreducibles of ℤ/4ℤ send the generator to 1, i, −1, −i. Over ℝ, the representations sending g to 1 and −1 remain irreducible (they are real). But the representations sending g to i and −i cannot be realized in dimension 1 over ℝ. They combine into a single 2-dimensional real irreducible representation where g acts as the rotation matrix [[0,−1],[1,0]]. This is the real 90° rotation representation.