Chapter 1
General results from representation theory

1.1. Basic definitions

Let G be a finite group. A representation of G on a vector space V is a pair (π,V ) where V is a complex vector space and π is a homomorphism G GL(V ). Often, we will denote (π,V ) simply by π, specially when the vector space V is specified implicitly. The dimension of V is called the degree of the representation (π,V ). In these notes all representations will be assumed to be of finite degree. If (π,V ) and (τ,U) are two representations of G, then a linear map φ : U V is called a homomorphism of G-modules, or an intertwiner if

φ(τ(g)u) = π(g)φ(u) for all u ∈ U.
The space of all homomorphisms (τ,U) (π,V ) will be denoted by HomG(τ,π). When φ is invertible, it is an isomorphism, and we say that τ is isomorphic to φ. The representations π and τ are said to be disjoint if HomG(τ,π) = 0.

1.2. The Pontryagin dual of a finite abelian group

Let G be an abelian group. The binary operation on the group will be written additively. A character of G is a homomorphism χ : G C*. In other words, χ(x + x) = χ(x)χ(x) for all x,x∈ G. A character χ is called unitary if χ(x)= 1 for all x ∈ G.

Exercise 1.1. Show that every character of a finite abelian group is unitary.

If G is a finite abelian group, its Pontryagin dual is the set ^G of its characters. Under point-wise multiplication of characters, ^
G forms a group. Once again, the binary operation is written additively, so that given characters χ and χof G, (χ + χ)(x) = χ(x)χ(x) for all x ∈ G. This is a special case of a general construction for locally compact abelian groups.

Proposition 1.2. For any finite abelian group G, G~
= ^

Proof. The proof is a sequence of exercises:

Exercise 1.3. Show that the Proposition is true for a finite cyclic group Z∕nZ.

Exercise 1.4. If G1 and G2 are abelian groups, show that

G1 × G2 ~= G^1 × ^G2.

Exercise 1.5. Show that every finite abelian group is isomorphic to a product of finite cyclic groups.

It follows from the above proposition that G^^~=G. However, in this case, there is a canonical isomorphism G ^^
G given by g↦→ǧ where ǧ is defined by

ˇg(χ) = χ(g) for each χ ∈G.

1.3. Induced Representations

Let H be a subgroup of G. Given a representation (π,V ) of H, the representation of G induced from π is the representation (πG,V G) where

V G = {f : G → V∣f(hg) = π(h)f(g) for all h ∈ H, g ∈ G}.
The action of G on such functions is by right translation
(πG(g)f)(x) = f(xg).

Now suppose that (τ,U) is a representation of G and (π,V ) is a representation of H. Because H G, we can regard U as a representation of H by restricting the homomorphism G GL(U) to H. Denote this representation by τH. Given φ ∈ HomG(τ,πG), define ~φ : U V by

~φ(u) = φ(u)(1) for each u ∈ U.

Exercise 1.6. Show that ~φ ∈ HomH(τH).

Theorem (Frobenius reciprocity). The map φ↦→~φ induces an isomorphism

Hom  (τ,πG) ~→ Hom  (τ  ,π).
    G            H  H

Proof. For ψ ∈ HomH(τH) define ψ~ : U V G by

~ψ(u)(x) = ψ(τ(x)u) for each u ∈ U and x ∈ G.

Exercise 1.7. For all h ∈ H, ψ~ (u)(hx) = π(h)~ψ (u)(x). Therefore, ψ~(u) ∈ V G.

Exercise 1.8. Show that ~ψ ∈ HomG(τ,πG).

Exercise 1.9. For all φ ∈ HomG(τ,πG), ~~φ = φ, and for all ψ ∈ HomH(τH), ~~
ψ = ψ.

Therefore the maps φ↦→~φ and ψ↦→~ψ are mutual inverses.

1.4. Description of intertwiners

In this section we describe the homomorphisms between two induced representations. Let G be a finite group. Let H1 and H2 be subgroups. Let (π1,V 1) and (π2,V 2) be representations of H1 and H2 respectively. For f : G V 1, and Δ : G HomC(V 1,V 2), define a convolution Δ * f : G V 2 by

(Δ * f)(x) = -1-   Δ(xg -1)f(g).
            ∣G ∣g∈G
Let D be the set of all functions Δ : G HomC(V 1,V 2) satisfying
Δ(h2gh1) = π2(h2)∘Δ(g) ∘π1(h1)
for all h1 ∈ H1, h2 ∈ H2 and g ∈ G.

Exercise 1.10. Show that if Δ ∈ D and f1 ∈ V 1G then Δ*f1 ∈ V 2G.

Exercise 1.11. Show that the map LΔ : V 1G V 2G defined by f1↦→Δ * f1 is a homomorphism of G-modules.

Theorem 1.12 (Mackey). The map Δ↦→LΔ is an isomorphism from D HomG(V 1G,V 2G).

Proof. We construct an inverse mapping HomG(V 1G,V 2G) D. For this, let us define a collection fg,v of elements in V 1G indexed by g ∈ G and v ∈ V 1:

        {π1(h)v   if x = hg,h ∈ H1
fg,v(x) =
        (0        if x ∕∈ H1g.

Exercise 1.13. Show that for every v ∈ V 1, we have

Δ(g)(v) = [G : H1]LΔ(fg-1,v)(1).

The above equation can be turned around to define, for each L : HomG(V 1G,V 2G) a function Δ ∈ D.

Exercise 1.14. Show that if L ∈ HomG(V 1,V 2), then the function Δ : G HomC(V 1,V 2) defined by

ΔL(g)(v) = [G : H1]L(fg-1,v)(1)
is in D.

Exercise 1.15. Check that the maps Δ↦→ΔL and L↦→LΔ are inverses of each other.

1.5. A criterion for irreducibility

Let G be a finite group, H a subgroup and (π,V ) a representation of H. The space V G can be decomposed into a direct sum

  G        ⊕
V   =   -1         VHx-1H,
      Hx  H∈H \G∕H
where V Hx-1H consists of functions G V supported on Hx-1H:
V  -1  = {f : Hx -1H → V ∣f(hx-1h′) = π(h)f(x -1h′) for all h,h′ ∈ H}.
 Hx  H
V Hx-1H is stable under the action of π. Let πHx-1H denote the resulting representation of H on V Hx-1H and let xπHx-1Hx denote the representation of H xHx-1 on V given by xπ(h) = π(x-1hx).

Exercise 1.16. Show that f↦→(h↦→f(x-1h)) defines an isomorphism of representations

π  -1  ~= (xπ      -1)H.
 Hx  H      H∩xHx

We have proved

Proposition 1.17. Let G be a finite group and H any subgroup. For every representation π of H, there is a canonical isomorphism of representations of H

(πG)H =              (xπH∩xHx- 1)H .
        Hx -1H∈H\G ∕H

By Frobenius reciprocity,

       G               G
EndG( π  ) =   HomH⊕((π )H,π)
           =               HomH  ((xπH ∩xHx-1)H ,π).
               Hx-1H∈H \G∕H
Recall that πG is irreducible if and only if EndG(πG) is one dimensional. As a result, we obtain Mackey’s irreducibility criterion:

Theorem (Mackey’s irreducibility criterion). Let G be a finite group and H a subgroup. Let π be an irreducible representation of H. Then πG is irreducible if and only if, for any x∕∈H, the representations π and (xπHxHx-1)H are disjoint.

Corollary 1.18. Suppose that G is a finite group and H a normal subgroup. Then for any irreducible representation π of H, πG is irreducible if and only if for every x∕∈H, xπ is not isomorphic to π.

1.6. The little groups method

The little groups method was first used by Wigner [Wig39], and generalized by Mackey [Mac58] to construct representations of a group from those of a normal subgroup. We will restrict ourselves to the case where G is a finite group and N is a normal subgroup of G which is abelian. Let N^ denote the Pontryagin dual of N (Section 1.2). Define an action of G on N^ by

g         -1                       ^
 χ(n) = χ(g ng) for each g ∈ G, χ ∈ N .
Let ρ be an irreducible representation of G on the vector space V ρ. For each χ ∈^N, write
Vχ = {x ∈ V ∣ρ(n)x = χ(n)x}.
Vρ =    Vχ.
N^(ρ) = {χ ∈ ^N ∣V ⁄= 0}.

Proposition 1.19 (Clifford’s theorem). N^(ρ) consists of a single G-orbit of ^

Proof. Suppose x ∈ V χ, and g ∈ G. Then

ρ(n)(ρ(g)x)  =  ρ(g)ρ(g  ng)x
           =  gχ ρ(g)x.
ρ(g)Vχ = Vgχ.

It follows that g∈GV gχ is invariant under ρ. From the irreducibility of ρ one concludes that if V χ⁄=0, then g∈GV gχ = V ρ.

For χ ∈N^(ρ), let

G χ = {g ∈ G ∣gχ = χ}.
It follows from (1.1) that for every g ∈ Gχ, ρ(g) preserves V χ. Therefore, ρ gives rise to a representation ρχ of Gχ on V χ.

Proposition 1.20 (Mackey’s imprimitivity theorem).

  ~  G
ρ = ρχ.


       ⊕      g
Vρ =         V χ.
    gGχ∈G ∕G χ
Therefore, for each x ∈ V ρ, there is a unique decomposition
x =         xgGχ.
   Gχg∈G χ\G
By (1.1), ρ(g-1)xgGχ ∈ V χ. The representation space of ρχG is
VG = {f : G → C ∣f (g′g) = χ(g′)f(g) for all g′ ∈ G χ g ∈ G}.
Define φ(x)(g) = ρ(g)xg-1Gχ for each g ∈ G.

Exercise 1.21. Show that φ : V ρ ρχG is a well defined isomorphism of representations of G.