Math 202B: Lecture 19

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In Lecture 17, we showed that for any two Hilbert spaces V and W the set \mathrm{Hom}(V,W) of linear maps T \colon V \to W is not just a vector space: it becomes a Hilbert space when equipped with the Frobenius scalar product \langle A,B \rangle = \mathrm{Tr}\, A^*B. Now suppose that (V,\varphi) and (W,\psi) are not just Hilbert spaces, but unitary representations of a group G. Then, the Hilbert space \mathrm{Hom}(V,W) is also a unitary representation of G in a natural way.

Definition 19.1 The adjoint representation of G corresponding to (V,\varphi) and (W,\psi) is (\mathrm{Hom}(V,W),\omega), where \omega = \omega(\varphi,\psi) is defined by

\omega(g)T = \psi(g)T\varphi(g)^*, \quad T \in \mathrm{Hom}(V,W),\ g \in G.

We need to check that Definition 18.5 really does specify a unitary representation of G. First, this means that \omega must take values in the unitary group of the Hilbert space \mathrm{Hom}(V,W), and we checked in Lecture 17 that for any unitary operators A \in \mathcal{L}(V) and B \in \mathcal{L}(W) the operator \omega(A,B) \in \mathcal{L}(\mathrm{Hom}(V,W)) defined by \omega(A,B)T=BTA^* is unitary, i.e. preserves the Frobenius scalar product. Second, we need to check that \omega is a group homomorphism. Let g,h \in G. Then, for any T \in \mathrm{Hom}(V,W), we have

\omega(gh)T = \psi(gh)T\varphi(gh)^* = \psi(g)\psi(h)T\varphi(h)^*\varphi(g)^*=\psi(g)[\omega(h)T]\varphi(g)^*=\omega(g)[\omega(h)T],

so indeed we have \omega(gh)=\omega(g)\omega(h). It is also clear that \omega(e)=I, with e \in G the group identity and I \in \mathcal{L}(\mathrm{Hom}(V,W)) the identity operator on transformations V \to W.

The notion of adjoint representation plays a key role in classifying unitary representations of G because it provides an alternative description of the G-linear maps between two possibly different unitary representations as the space of G-invariants in a single unitary representation. We now explain this in more detail.

Definition 19.1. For any unitary representation (V,\varphi) of G, the G-invariant subspace of V is

V^G = \{ v \in V \colon \varphi(g)v=v\text{ for all }g \in G\}.

In other words, V^G is the set of all vectors v\in V which are fixed points of all the operators \varphi(g), g \in G.

Proposition 19.1. The space V^G is a vector subspace of V, invariant under all of the operators \varphi(g), g \in G. In other words, V^G is a subrepresentation of V.

Problem 19.1. Prove Proposition 18.1.

We now come to the key point, which is that the space \mathrm{Hom}_G(V,W) of G-linear maps between two unitary representations (V,\varphi) and (W,\psi) of G is the same thing as the space of G-invariants in the corresponding adjoint representation (\mathrm{Hom}(V,W),\omega).

Proposition 19.2. \mathrm{Hom}(V,W)^G = \mathrm{Hom}_G(V,W).

Proof: We have

\mathrm{Hom}(V,W)^G = \{T \in \mathrm{Hom}(V,W) \colon \omega(g)T =T \text{ for all }g \in G\} \\ = \{T \in \mathrm{Hom}(V,W) \colon \psi(g)T\varphi(g)^* =T \text{ for all }g \in G\} \\ = \{T \in \mathrm{Hom}(V,W) \colon \psi(g)T =T\varphi(g) \text{ for all }g \in G\}.

-QED

Proposition 18.2 would be just a curiosity were it not for the fact that there is an explicit way to access the G-invariants of a unitary representation.

Theorem 19.2. (First Projection Formula) For any unitary representation (V,\varphi) of G, the operator P^G \in \mathcal{L}(V) defined by averaging all operators in the representation,

P^G = \frac{1}{|G|} \sum\limits_{g \in G} \varphi(g),

is a projection in \mathcal{L}(V) whose image is V^G.

Problem 19.2. Prove Theorem 18.2.

We will apply the First Projection Formula to the adjoint representation in a moment. First, we want to point out another algebraic application of averaging in the context of group representations. One might ask why we do not consider a more general notion of representation for a given group G, in which we declare a representation to be any pair (V,\varphi) consisting of a vector space together with a group homomorphism \varphi \colon G \to \mathrm{GL}(V) into the group of all invertible operators on V. It is an interesting fact that this ostensible more general construction is in fact not more general. To see why, start with any initial scalar product \langle \cdot,\cdot \rangle_0 on the vector space V, and define a new scalar product on V by

\langle v,w \rangle = \frac{1}{|G|}\sum\limits_{g \in G} \langle \varphi(g)v,\varphi(g)w \rangle_0.

In other words, the new scalar product \langle v,w \rangle of two given vectors is the expected value of the old scalar product \langle \varphi(g)v,\varphi(g)w\rangle_0 of these vectors after each is randomly rotated \rho(g), where g is a uniformly random element of the finite group G.

We now return to the adjoint representation (\mathrm{Hom}(V,W),\omega) corresponding to two given unitary representations (V,\varphi) and (V,\psi) of the group G. On one hand, we have

\dim \mathrm{Hom}(V,W)^G = \dim \mathrm{Hom}_G(V,W)

as a direct consequence of Proposition 18.2. On the other hand, by the First Projection Formula the operator

P = \frac{1}{|G|} \sum\limits_{g \in G} \omega(g)

is the orthogonal projection of \mathrm{Hom}(V,W) onto the subspace \mathrm{Hom}(V,W)^G = \mathrm{Hom}_G(V,W), so that in particular

\mathrm{Tr}\, P = \dim \mathrm{Hom}(V,W)^G = \dim \mathrm{Hom}_G(V,W).

This is because, as you know from Math 202A, the trace of a projection is the dimension of its image. We want to compute this trace. Since

\mathrm{Tr}P = \frac{1}{|G|} \sum\limits_{g \in G} \mathrm{Tr} \omega(g),

the real issue is to compute the trace of \omega(g) for each g \in G. This is a good time to recall that \omega(g)T = \psi(g)T\varphi(g)^* for all T \in \mathrm{Hom}(V,W).

Let Y \subset V and X \subset W be orthonormal basis, and let \{E_{xy} \colon (x,y) \in X \times Y\} be the corresponding orthonormal basis of the Hilbert space \mathrm{Hom}(V,W) as in Lecture 18. Also in Lecture 18, we showed that for any A \in \mathcal{L}(V) and B \in \mathcal{L}(W), the matrix elements of \omega(A,B) \in \mathcal{L}(\mathrm{Hom}(V,W)) are given by the product formula

\langle E_{xy},\omega(A,B)E_{x'y'} \rangle = \overline{\langle y,Ay'\rangle_V}\langle x,Bx'\rangle_W.

Thus, we have

\mathrm{Tr} \omega(g) = \sum\limits_{x \in X} \sum\limits_{y \in Y} \langle E_{xy},\omega(\varphi(g),\psi(g))E_{xy}\rangle=\sum\limits_{x \in X} \sum\limits_{y \in Y} \overline{\langle y,\varphi(g)y\rangle_V}\langle x,\psi(g)x\rangle_W = \overline{\mathrm{Tr} \varphi(g)} \mathrm{Tr} \psi(g).

Definition 19.2. For any unitary representation (V,\varphi) of G, the function \chi^\varphi \colon G \to \mathbb{C} defined by \chi^\varphi(g) = \mathrm{Tr}\varphi(g) is called the character of (V,\varphi).

Problem 19.3. Prove that |\chi^\varphi(g)| \leq \dim V for all g \in G.

In terms of characters, the calculation we performed above can be stated simply as

\chi^\omega(g) = \overline{\chi^\varphi(g)}\chi^\psi(g), \quad g \in G.

The trace of the orthogonal projection

P=\frac{1}{|G|} \sum\limits_{g \in G} \omega(g)

of the adjoint representation \mathrm{Hom}(V,W) onto the subspace \mathrm{Hom}(V,W)^G=\mathrm{Hom}_G(V,W) is

\mathrm{Tr}P = \frac{1}{|G|} \sum\limits_{g \in G} \overline{\chi^\rho(g)} \chi^\psi(g) = \frac{1}{|G|} \langle \chi^\rho,\chi^\psi\rangle_2,

where \langle \cdot,\cdot\rangle_2 is the \ell^2 scalar product on the vector space \mathcal{F}(G)=\mathcal{C}(G) of complex-valued functions on the group G. We have thus proved the following Theorem.

Theorem 19.3. For any two unitary representations (V,\varphi) and (W,\psi) of G, we have

\dim \mathrm{Hom}_G(V,W) = \frac{1}{|G|}\langle \chi^\rho,\chi^\psi\rangle_2.

In Lecture 19, we will combine this result with Schur’s Lemma to classify irreducible unitary representations of a finite group G up to isomorphism.

Published by Jonathan Novak

Mathematician at UC San Diego.

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