General results from representation theory

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

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.

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 χ′ of 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.

Proof. The proof is a sequence of exercises:

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 gǧ where ǧ is defined by

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

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 τ_{H}. Given φ Hom_{G}(τ,π^{G}), define : U → V
by

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

Proof. For ψ Hom_{H}(τ_{H},π) define : U → V ^{G} by

Therefore the maps φ and ψ are mutual inverses. □

In this section we describe the homomorphisms between two induced
representations. Let G be a finite group. Let H_{1} and H_{2} be subgroups. Let
(π_{1},V _{1}) and (π_{2},V _{2}) be representations of H_{1} and H_{2} respectively. For
f : G → V _{1}, and Δ : G → Hom_{C}(V _{1},V _{2}), define a convolution Δ * f : G → V _{2}
by

Exercise 1.11. Show that the map L_{Δ} : V _{1}^{G} → V _{2}^{G} defined by
f_{1}Δ * f_{1} is a homomorphism of G-modules.

Proof. We construct an inverse mapping Hom_{G}(V _{1}^{G},V _{2}^{G}) → D. For
this, let us define a collection f_{g,v} of elements in V _{1}^{G} indexed by g G and
v V _{1}:

The above equation can be turned around to define, for each
L : Hom_{G}(V _{1}^{G},V _{2}^{G}) a function Δ D.

Exercise 1.14. Show that if L Hom_{G}(V _{1},V _{2}), then the function
Δ : G → Hom_{C}(V _{1},V _{2}) defined by

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

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

Theorem (Mackey’s irreducibility criterion). Let G be a finite
group and H a subgroup. Let π be an irreducible representation of H.
Then π^{G} is irreducible if and only if, for any xH, the representations
π and (^{x}π_{H∩xHx-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, π^{G} is
irreducible if and only if for every xH, ^{x}π is not isomorphic to π.

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

Proof. Suppose x V _{χ}, and g G. Then

| (1.1) |

It follows that ⊕_{gG}V _{gχ} is invariant under ρ. From the irreducibility of ρ one
concludes that if V _{χ}0, then ⊕_{gG}V _{gχ} = V _{ρ}. □

For χ (ρ), let

Proof.

Exercise 1.21. Show that φ : V _{ρ} → ρ_{χ}^{G} is a well defined
isomorphism of representations of G.