Math 202B: Lecture 17

Let \mathcal{A} = \mathcal{C}(G) be the convolution algebra of a finite group G, and let \rho \colon \mathcal{A} \to \mathcal{L}(V) be a linear map from \mathcal{A} into the algebra of linear operators on a Hilbert space V. We define an associated function on the underlying group G by

U^\rho(g) = \rho(E_g), \quad g \in G,

where E_g(h) = \delta_{gh} is the elementary function on G indexed by the group element g \in G. By definition, U^\rho is a function with domain G and codomain \mathcal{L}(V).

Theorem 17.1. The linear map \rho \colon \mathcal{C}(G) \to \mathcal{L}(V) is an algebra homomorphism if and only if U^\rho is a group homomorphism from G into U(\mathcal{L}(V)), the unitary group of the algebra \mathcal{L}(V).

Proof: Suppose first that \rho is an algebra homomorphism from \mathcal{C}(G) to \mathcal{L}(V). Then, for any g,h \in G we have

U^\rho(gh) = \rho(E_{gh}) = \rho(E_gE_h) = \rho(E_g)\rho(E_h) = U^\rho(g)U^\rho(h),

and moreover for e \in G the group unit

U^\rho(e) =\rho(E_e) = I,

where I \in \mathcal{L}(V) is the identity operator on V. Consequently, we have

U^\rho(g) U^\rho(g^{-1}) = U^\rho(gg^{-1}) = U^\rho(e)=I,

which shows that U^\rho(g) is invertible in \mathcal{L}(V) with inverse U^\rho(g)^{-1}=U^\rho(g^{-1}). Thus U^\rho is a group homomorphism from G into the group of invertible elements in \mathcal{L}(V). It remains to show that the codomain of U^\rho is in fact the smaller group U(\mathcal{L}(V)). This is verified by

U^\rho(g)^*=\rho(E_g)^*=\rho(E_g^*)=\rho(E_{g^{-1}}) = U^\rho(g^{-1})=U^\rho(g)^{-1}.

Conversely, suppose we start from the assumption that U^\rho is a group homomorphism from G into the unitary group U(\mathcal{L}(V)). Note that \rho is a linear map by hypothesis; what we have to check is that it respects multiplication and conjugation in the convolution algebra \mathcal{C}(G), and maps E_e to I. It is sufficient to check this for the elementary basis, and we have

\rho(E_gE_h) = \rho(E_{gh})=U^\rho(gh)=U^\rho(gh) = U^\rho(g)U^\rho(h)=\rho(E_g)\rho(E_h),

and also

\rho(E_e) = U^\rho(e)=I,

where in both calculations we are using the hypothesis that U^\rho is a group homomorphism. Finally,

\rho(E_g^*)=\rho(E_{g^{-1}}) = U^\rho(g^{-1})=U^\rho(g)^{-1}=U^\rho(g)^*=\rho(E_g)^*,

where we used our hypothesis that the codomain of the group homomorphism U^\rho is the unitary group U(\mathcal{L}(V)).

-QED

You will recognize that Theorem 17.1 was proved previously in the special case where V is a one-dimensional Hilbert space, so that \mathcal{L}(V) \simeq \mathbb{C} is the algebra of complex numbers and U(\mathcal{L}(V)) is the unit circle in \mathbb{C}. In this one-dimensional setting, we gathered enough information to construct the Fourier transform on \mathcal{C}(G) for G abelian. For G nonabelian, our path is the same except that we have to consider homomorphisms from G into the unitary group of higher-dimensional Hilbert spaces V.

Definition 17.1. A unitary representation of G is a pair (V,U) consisting of a Hilbert space V together with a group homomorphism

U\colon G \longrightarrow U(\mathcal{L}(V)),

where U(\mathcal{L}(V)) is the unitary group of the algebra of all linear operators on V.

By Theorem 17.1, there is a bijective correspondence between linear representations of the convolution algebra \mathcal{C}(G) and unitary representations of the underlying group G. Therefore, in the case of convolution algebras, we can and do choose to work with unitary representations of G. All the basic notions we need about these object essentially coincide with those already developed in the more general setting of linear representations of algebras.

Definition 17.2. If (V_1,U_1) and (V_2,U_2) are unitary representations of G. A homomorphism of unitary representations is a linear map T \colon V_1 \to V_2 such that

T \circ U_1(g) =U_2(g) \circ T, \quad \text{for all }g \in G.

The set of all such linear maps is denoted \mathrm{Hom}_G(V_1,V_2). If T as above is a vector space isomorphism, then we say that (V_1,U_1) and (V_2,U_2) are isomorphic unitary representations of the group G.

Theorem 17.2. Suppose that (V_1,U_1) and (V_2,U_2) are isomorphic unitary representations of G, i.e. there exists a linear isomorphism in \mathrm{Hom}_G(V_1,V_2). Then, a stronger statement holds: there exists a unitary isomorphism T \in \mathrm{Hom}_G(V_1,V_2), i.e. an isometric isomorphism. To be completely explicit, there is T \in \mathrm{Hom}_G(V_1,V_2) such that

\langle Tv_1,Tw_1\rangle = \langle v_1,w_1\rangle, \quad \text{for all } v_1,w_1 \in V_1.

Proof: The proof is an (interesting) exercise in polar decomposition, or singular value decomposition if you prefer. -QED

To further connect up with linear representations of algebras, let (V,\rho) be a linear representation of \mathcal{C}(G), and let (V,U^\rho) be the corresponding unitary representation of G, as in Theorem 17.1

Theorem 17.3. The linear representation (V,\rho) is irreducible if and only if the unitary representation (V,U^\rho) is irreducible.

Problem 17.1. Prove Theorem 17.3. (This is straightforward and is just to get you accustomed to the definitions in play).

Theorem 17.4 (Schur’s Lemma) Let (V_1,U_1) and (V_2,U_2) be unitary representations of G. If (V_1,U_1) is irreducible, every homomorphism T \in \mathrm{Hom}_G(V_1,V_2) is injective. If (V_2,U_2) is irreducible, every T \in \mathrm{Hom}_G(V_1,V_2) is surjective.

Proof: Special case of Schur’s lemma for linear representations of algebras. -QED

Theorem 17.5. (Maschke’s Theorem) Let (V,U) be a unitary representation of G. Then, there exists a linear partition

V = \bigoplus\limits_{W \in \mathsf{W}} W

of V whose such that, for every W \in \mathsf{W}, the unitary representation (W,U_W) obtained by restricting each U(g) to W is irreducible. That is, every unitary representation decomposes into a direct sum of irreducible unitary representations.

Proof: Special case of Maschke’s theorem for linear representations of algebras.

-QED

Our goal is to classify all unitary representations of a finite group G, up to isomorphism. By Maschke’s theorem, it suffices to classify the irreducible ones (“irreps”). If G is abelian, this is the classification of homomorphisms G \to U(\mathbb{C}), which we successfully achieved in our development of the Fourier transform; we called the parameterization set \Lambda, and it is an explicit set of tuples of roots of unity corresponding to the decomposition of G into a product of cyclic groups.

Now suppose that G is a nonabelian group, and let \Lambda be a set parameterizing isomorphism classes of irreducible unitary representations of \group{G}. We know absolutely nothing about \Lambda yet, but still we can get going. For each \lambda \in \Lambda, let (V^\lambda,U^\lambda) be a representative of the corresponding isomorphism class of irreducible unitary representations. Let X^\lambda \subset V^\lambda be an orthonormal basis, and for each x,y \in X^\lambda

\mu_{xy}^\lambda \colon G \longrightarrow \mathbb{C}

be the function on G defined by

\mu_{xy}^\lambda(g) = \langle x,U^\lambda(g)y\rangle.

One of the two main theorems next week is the following.

Theorem 17.6. The set \{\mu_{xy}^\lambda \colon \lambda \in \Lambda,\ x,y \in X^\lambda\} is an orthogonal basis of the convolution algebra \mathcal{C}(G).

This theorem generalizes the construction of the Fourier basis in \mathcal{C}(G) in the case where G is abelian. In fact, it is not too difficult to prove – the hardest part is to make sure you absorb and understand the definitions needed to state it. If you can do this, the logic of the argument is not difficult to follow.

Published by Jonathan Novak

Mathematician at UC San Diego.

Leave a Reply