Math 202B: Lecture 21

***Problems in this lecture due March 8***

Let G be a finite group. The notion of adjoint representation plays a key role in understanding the unitary representation category \mathbf{Rep}(G) because it leads to a dimension formula for its hom-spaces, \mathrm{Hom}_G(V,W). We now explain this.

Definition 21.1. For any unitary representation (V,\varphi) of G, the space of G-invariant vectors in 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 each unitary operator \varphi(g), g \in G.

Proposition 21.2. The space V^G is a vector subspace of V, which is moreover mapped into itself by each of the operators \varphi(g), $g \in G.$

Problem 21.1. Prove Proposition 21.2.

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

Proposition 21.3. \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 21.3 reduces the problem of describing \mathrm{Hom}_G(V,W) to the problem of describing \mathrm{Hom}(V,W)^G. This is progress, because the problem of describing V^G can be solved for any unitary representation (V,\varphi) of G.

Theorem 21.4. (First Projection Formula) For any unitary representation (V,\varphi) of G, the operator P \in \mathrm{End}(V) defined by averaging all operators in the representation,

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

is a selfadjoint projections whose image is V^G.

Problem 21.2. Prove Theorem 21.4, which is our third application of averaging over a group.

We will apply the First Projection Formula to the adjoint representation (\mathrm{Hom}(V,W),\omega) corresponding to two given unitary representations (V,\varphi) and (V,\psi) of 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)=\frac{1}{|G|} \sum\limits_{g \in G}\chi^{\mathrm{Hom}(V,W)}(g),

and since we showed in Lecture 20 that

\chi^{\mathrm{Hom}(V,W)}(g)=\overline{\chi^V(g)}\chi^W(g),

we have established the following.

Theorem 21.5. 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^V,\chi^W\rangle,

where \langle \cdot,\cdot \rangle is the L^2-scalar product on \mathcal{C}(G).

Theorem 21.5 opens up a path to establishing a version of character orthogonality in \mathcal{C}(G). Namely, we want to build a special subcategory of \mathrm{Rep}(G) such that if (V,\rho) and (W,\psi) are distinct objects in this subcategory then the space \mathrm{Hom}_G(V,W) of morphisms between them is the zero space. We will see how to do this next week using a structural (as opposed to numerical) description of \mathrm{Hom}_G(V,W) based on the notion of irreducible representations – those which cannot be be decomposed into smaller ones.

Published by Jonathan Novak

Mathematician at UC San Diego.

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