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:
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 (πG,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 τH. Given φ HomG(τ,πG), define : U → V by
Theorem (Frobenius reciprocity). The map φ induces an isomorphism
Proof. For ψ HomH(τH,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 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
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:
The above equation can be turned around to define, for each L : HomG(V 1G,V 2G) a function Δ D.
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
By Frobenius reciprocity,
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 xH, the representations π and (xπH∩xHx-1)H are disjoint.
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
It follows that ⊕gGV gχ is invariant under ρ. From the irreducibility of ρ one concludes that if V χ0, then ⊕gGV gχ = V ρ. □
For χ (ρ), let