## Chapter 1General 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 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 of its characters. Under point-wise multiplication of characters, 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 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. However, in this case, there is a canonical isomorphism G given by g ǧ where ǧ is defined by ### 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 The action of G on such functions is by right translation 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 : U V by Exercise 1.6. Show that  HomH(τH).

Theorem (Frobenius reciprocity). The map φ  induces an isomorphism Proof. For ψ HomH(τH) define : U V G by Exercise 1.7. For all h H, (u)(hx) = π(h) (u)(x). Therefore, (u) V G.

Exercise 1.8. Show that  HomG(τ,πG).

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

Therefore the maps φ  and 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 Let D be the set of all functions Δ : G HomC(V 1,V 2) satisfying 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: Exercise 1.13. Show that for every v V 1, we have 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 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 where V Hx-1H consists of functions G V supported on Hx-1H: 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 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  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 denote the Pontryagin dual of N (Section 1.2). Define an action of G on by Let ρ be an irreducible representation of G on the vector space V ρ. For each χ  , write Then Define Proposition 1.19 (Clifford’s theorem). (ρ) consists of a single G-orbit of .

Proof. Suppose x V χ, and g G. Then Therefore, (1.1)

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  (ρ), let 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). Proof. Therefore, for each x V ρ, there is a unique decomposition By (1.1), ρ(g-1)xgGχ V χ. The representation space of ρχG is 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.