Questions: Induced Representations

Question 1

If H has index 3 in G and σ is a 2-dimensional representation of H, what is the dimension of Ind_H^G(σ)?

• A. 2
• B. 3
• C. 5
• D. 6

Answer: (D)

dim(Ind_H^G(σ)) = [G:H] · dim(σ) = 3 · 2 = 6. The induced representation is built from [G:H] copies of the representation space, one for each left coset of H in G. Each copy is 'rotated' into the others by elements of G not in H, creating a larger space on which G acts.

Question 2

The character of an induced representation Ind_H^G(σ) is given by the formula χ^G(g) = (1/|H|) Σ_{x∈G, x⁻¹gx∈H} χ_σ(x⁻¹gx). Why does the sum only include x with x⁻¹gx ∈ H?

Model answer: The character formula involves extending χ_σ to all of G by setting it to zero outside H. The term χ_σ(x⁻¹gx) is only nonzero when x⁻¹gx ∈ H, so only those terms contribute. Geometrically, we are summing over the coset representatives x that conjugate g back into H, where σ can 'see' the element.

This formula shows that induction depends on how the conjugacy classes of G intersect H. If g is conjugate to no element of H, the induced character is zero at g. The formula can also be written as a sum over coset representatives, making the coset structure of G/H explicit.

Question 3

Restriction and induction are inverse operations: inducing and then restricting always returns the original representation.

• T. True
• F. False

Answer: False

Restriction and induction are adjoint functors (this is Frobenius reciprocity), not inverse operations. Res_H^G(Ind_H^G(σ)) is generally much larger than σ — it contains σ as a summand but also contains other components arising from how the cosets interact with H. The precise relationship is given by Mackey's formula.

Question 4

Inducing the trivial representation of H to G gives a representation whose character counts what?

• A. The number of elements in each conjugacy class
• B. The number of fixed points of g acting on G/H (the coset space)
• C. The order of the centralizer of each element
• D. The number of subgroups conjugate to H

Answer: (B)

The induced character from the trivial representation of H is χ(g) = |{xH ∈ G/H : gxH = xH}|, which counts the number of cosets fixed by g. This is the permutation character of the action of G on G/H. When H = {e}, this gives the regular representation. When H is larger, it gives the representation associated to the natural action of G on the coset space.