Math 202B: Lecture 19

We now understand the convolution algebra \mathcal{C}(G) very well for

G=G_1 \times \dots \times G_r

be a product group with G_i a cyclic group of order n_i. The basic mechanism is that we can parameterize both the group G=\{g_\alpha \colon \mu \in \Lambda\} and the set \widehat{G}=\{\chi^\lambda \colon \lambda \in \Lambda\} of multiplicative characters of G by

\Lambda = \mathbb{Z}_{n_1} \times \dots \times \dots \mathbb{Z}_{n_r}

in a very explicit and useful way. Namely, for every \alpha,\lambda \in \Lambda we have

\chi^\lambda(g_\alpha) = \omega_1^{\alpha_1\lambda_1} \dots \omega_r^{\alpha_r\lambda_r}, \quad \omega_k=\exp\left(\frac{2\pi i}{n_k}\right),

and this is the complete list of homomorphisms G \to U(1) from G into the unit circle U(1) in \mathbb{C}. Using this description, we were able to show that \widehat{G}=\{\chi^\lambda \in \lambda \in \Lambda\} is an orthogonal basis of \mathcal{C}(G),

\langle \chi^\lambda,\chi^\mu\rangle = \sum\limits_{g \in G} \overline{\chi^\lambda(g)}\chi^\mu(g)=\delta_{\lambda\mu}|G|.

Moreover, the normalized characters

F^\lambda :=\frac{1}{|G|}\chi^\lambda, \quad \lambda \in \Lambda,

form a Fourier basis \{F^\lambda \colon \lambda \in \Lambda\} of \mathcal{C}(G). Thus we get an algebra isomorphism \mathcal{C}(G) \to \mathcal{F}(\Lambda), the Fourier transform.

Now let us compare the Fourier transform on \mathcal{C}(G) with another kind of “transform” we have seen already, the regular representation of an algebra \mathcal{A} equipped with a Frobenius scalar product \langle \cdot,\cdot \rangle, i.e. a von Neumann algebra). In this setting we looked at the algebra homomorphism

\Phi \colon \mathcal{A} \longrightarrow \mathrm{End}\mathcal{A}

which sends each algebra element A \in \mathcal{A} to the left multiplication operator

\Phi(A)B=AB, \quad B \in \mathcal{A}.

Specializing this construction to the case where \mathcal{A}=\mathcal{C}(G) is the convolution algebra of a finite abelian group G, we obtain the following spectral interpretation of the Fourier transform.

Theorem 19.1. For every A \in \mathcal{C}(G), the Fourier basis \{F^\lambda \colon \lambda \in \Lambda\} is an eigenbasis of \Phi(A). More precisely, we have

\Phi(A)F^\lambda = \widehat{A}(\lambda)F^\lambda, \quad \lambda \in \Lambda.

Proof: We have

\Phi(A)F^\lambda = \left(\sum\limits_{\mu \in \Lambda} \widehat{A}(\mu)F^\mu\right)F^\lambda = \sum\limits_{\mu \in \Lambda} \widehat{A}(\mu)F^\mu F^\lambda = \widehat{A}(\lambda)F^\lambda,

where we used the fact that F^\lambda F^\mu = \delta_{\lambda\mu}F^\lambda. \square

Problem 19.1. Explicitly calculate the eigenvalues and eigenvectors of circulant matrices. Use the uncertainty principle to relate the sparsity (number of nonzero entries) and nullity (number of zero eigenvalues) of circulant matrices.

Since every finite abelian group is isomorphic to a product of cyclic groups, we understand \mathcal{C}(G) very well for arbitrary finite abelian groups G. On the other hand, we have seen that there are noncommutative groups which only admit a single “trivial” character sending every element to 1. Thus, we cannot expect the multiplicative characters of G to provide us with such a complete theory of \mathcal{C}(G) for nonabelian groups G.

We now take a leap of faith and generalize the character concept by looking at homomorphisms from G into “noncommutative circles.” You might wonder how someone would make such an inspired guess, and the answer is that there was no particular “someone” who did. Rather, over time the work of many people gradually made it clear that this would be a good approach to develop. You can read more about this here. The resulting theory has now been gone over many times by successive generalizations of algebraists (any analysts, and physicists), resulting in a very elegant and streamlined flow of ideas which we will follow in our remaining lectures.

Definition 19.2. A unitary representation of G is a pair (V,\varphi) consisting of a Hilbert space V together with a group homomorphism \varphi \colon G \to U(V), where U(V) is the unitary group of V.

If we take V=\mathbb{C}, a group homomorphism \varphi \colon G \to \mathbb{U}(V) is the same thing as a multiplicative character of G, so unitary representations are indeed generalizations of multiplicative characters. Note that Definition 19.1 makes sense with G being any group, whether finite or infinite, and V being any Hilbert space, whether finite or infinite dimensional. However, in Math 202B we will work exclusively with finite-dimensional unitary representations of finite groups.

The basic difference between multiplicative characters \chi \colon G \to U(1) and unitary representations \varphi \colon G \to U(V) is that \chi is a scalar-valued function on G whereas \varphi is an operator-valued function on G. Thus, \chi \in \mathcal{C}(G) but \varphi \not\in \mathcal{C}(G). However, the operator \varphi(g) can be described by a matrix and as g varies over G this gives us a collection of functions in \mathcal{C}(G).

Definition 19.3. Let (V,\varphi) be a unitary representation of G and let X \subset V be an orthonormal basis of V. We then have a corresponding set of functions \{\mu_{yx}^V \colon (x,y) \in X \times X\} in \mathcal{C}(G) defined by

\mu_{yx}^V(g) = \langle y,\varphi(g)x\rangle, \quad g \in G.

The functions \mu_{xy}^V \in \mathcal{C}(G) are called the matrix elements of the (V,\varphi) relative to the basis X.

The matrix elements of (V,\varphi) depend on the chosen basis X \subset V, but as we know a particular combination of them does not.

Definition 19.3. The character of a unitary representation (V,\varphi) of G is the function \chi^V \in \mathcal{C}(G) defined by

\chi^V(g) =\mathrm{Tr}\ \varphi(g), \quad g \in G.

Here is an example of a unitary representation (V,\varphi) of a group G. We take V=\mathcal{C}(G)=\mathcal{F}(G) to be the space of complex-valued functions on G with its L^2-scalar product. We define a group homomorphism \varphi \colon G \to U(V) by

\varphi(g)A = E_gA, \quad A \in \mathcal{C}(G),

where E_g is the elementary function corresponding to g \in G. Then, \varphi(g) is indeed a linear operator on V, since

\varphi(g)(\alpha A +\beta B) = \alpha E_gA +\beta E_gB=\alpha \varphi(g)A + \beta \varphi(g)B.

Moreover, the linear operator \varphi(g) \in \mathrm{End}\mathcal{C}(G) is unitary because it permutes the elements of the orthonormal basis \{E_g \colon g \in G\},

\varphi(g)E_h = E_gE_g = E_{gh}, \quad h \in G.

Thus, the matrix of \varphi(g) relative to the elementary basis \{E_g \colon g \in G\} is not just a unitary matrix, it is a permutation matrix. This unitary representation is called the regular representation of G, and it is “the same thing” as the regular representation of \mathcal{C}(G) discussed above in the following sense: we have

\Phi(E_g) = \varphi(g), \quad g \in G.

Problem 19.2. Calculate the character of the regular representation of G.

For an arbitrary algebra \mathcal{A}, a linear representation of \mathcal{A} is a pair (V,\varphi) consisting of a Hilbert space V together with an algebra homomorphism

\Phi \colon \mathcal{A} \longrightarrow \mathrm{End}(V).

In the special case where \mathcal{A}=\mathcal{C}(G), every linear representation (V,\Phi) of \mathcal{C}(G) gives a unitary representation (V,\varphi) of G defined by

\varphi(g)=\Phi(E_g),\quad g \in G.

Conversely, every unitary representation (V,\varphi) of G gives a linear representation (V,\Phi) of \mathcal{C}(G) defined by

\Phi(E_g) = \varphi(g), \quad g \in G.

So, for convolution algebras \mathcal{C}(G), the study of linear representations of \mathcal{C}(G) is equivalent to the study of unitary representations of G.

One could drop the “unitary” clause and consider a more general construction in which we define a representation of G to be a pair (V,\rho) consisting of a Hilbert space V together with a group homomorphism

\varphi \colon G \longrightarrow GL(V)

from G into the group of all invertible linear transformations V \to V. Thus, we do not require \varphi(g) \in GL(V) to preserve the scalar product \langle \cdot,\cdot \rangle on V. We will only consider unitary representations, and it turns out that there is in fact no loss of generality here.

Problem 19.3. Given a representation (V,\varphi) of G, define a new scalar product on V by

\langle v,w \rangle_G = \frac{1}{|G|}\sum\limits_{g \in G} \langle \varphi(g)v,\varphi(g)w\rangle.

Show that each of the operators \varphi(g), g \in G, is unitary with respect to this new scalar product, i.e.

\langle \varphi(g)v,\varphi(g)w\rangle_G = \langle v,w\rangle_G, \quad v,w \in V.

Problem 19.3 is our second use of the averaging trick, and we will see a third application of this technique very soon. In problem 19.3, it is not important that V is finite-dimensional, but it is significant that G is finite. This result does hold for certain infinite groups called compact groups, but for noncompact groups unitary representations actually are a special subclass of representations.

Published by Jonathan Novak

Mathematician at UC San Diego.

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