Math 202B: Lecture 21

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Let \Lambda be a set parameterizing the isomorphism classes of irreducible unitary representations of a finite group G. For each \lambda \in \Lambda, let (V^\lambda,\rho^\lambda) be a representative of the isomorphism class of irreps corresponding to \lambda. The character of (V^\lambda,\rho^\lambda) is the central function \chi^\lambda \colon G \to \mathbb{C} defined by

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

Last time, we proved that \{\chi^\lambda \colon \lambda \in \Lambda\} is an orthogonal set in the class algebra \mathcal{Z}(G), which is the center of the convolution algebra \mathcal{C}(G). Specifically, we have

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

This tells us that |\Lambda| \leq \dim \mathcal{Z}(G), i.e. that the number of irreducible unitary representations of G (up to isomorphism) is at most the number of conjugacy classes in G. Today, we will develop additional consequences of character orthogonality.

Let (V,\varphi) be any unitary representation of G, not necessarily irreducible. Let (V,\Phi) be the corresponding linear representation of \mathcal{C}(G),

\Phi(A) = \Phi\left(\sum\limits_{g\in G} \alpha_g E_g \right) = \sum\limits_{g \in G} \alpha_g \varphi(g).

Let \mathcal{A} be the image of the convolution algebra \mathcal{C}(G) under \Phi. This is a subalgebra of the linear algebra \mathcal{L}(V), and by Maschke’s theorem there is a linear partition \mathbf{W} of V,

V = \bigoplus_{W \in \mathbf{W} } W,

whose blocks are \mathcal{A}-invariant and \mathcal{A}-irreducible subspaces. Each block W \in \mathbf{W} may thus be regarded as a unitary representation (W,\varphi^W) of G, where \varphi^W(g) \in U(\mathcal{L}(W)) is the restriction of \varphi(g) \in U(\mathcal{L}(V)) to the subspace W \leq V. Since (W,\varphi^W) is an irreducible unitary representation of G, it must be isomorphic to (V^\lambda,\rho^\lambda) for some \lambda \in \Lambda. Thus, the linear partition of V into \mathcal{A}-invariant, \mathcal{A}-irreducible subpsaces yields an isomorphism of unitary representations

V \simeq \bigoplus\limits_{\lambda \in \Lambda} \mathrm{Mult}_{\mathbf{W}}(V^\lambda,V) V^\lambda,

where \mathrm{Mult}_{\mathbf{W}}(V^\lambda,V) is the number of blocks of the linear partition \mathbf{W} which are isomorphic to V^\lambda, and

\mathrm{Mult}_{\mathbf{W}}(V^\lambda,V) V^\lambda = \underbrace{V^\lambda \oplus \dots \oplus V^\lambda}_{\mathrm{Mult}_{\mathbf{W}}(V^\lambda,V)\text{ times}}.

Theorem 21.1. We have

\mathrm{Mult}_{\mathbf{W}}(V^\lambda,V) =\frac{1}{|G|} \langle \chi^\lambda,\chi^V\rangle.

Note that the right hand side has no dependence on the linear partition \mathbf{W}.

Proof: Since isomorphic representations have the same character (proved in Lecture 20), we have

\chi^V = \sum\limits_{\mu \in \Lambda} \mathrm{Mult}_{\mathbf{W}}(V^\lambda,V)\chi^\mu,

which is an equality of functions in \mathcal{Z}(G). Thus, for any \lambda \in \Lambda we have

\langle \chi^\lambda,\chi^V \rangle = \sum\limits_{\mu \in \Lambda} \mathrm{Mult}_{\mathbf{W}}(V^\lambda,V) \chi^\lambda,\chi^\mu\rangle = |G| \mathrm{Mult}_{\mathbf{W}}(V^\lambda,V),

where the second equality follows from \langle \chi^\lambda,\chi^\mu\rangle = \delta_{\lambda\mu}|G|.

-QED

As a consequence of Theorem 21.1, we can make the following important definition.

Definition 21.1. The isotypic decomposition of a unitary representation (V,\varphi) of a finite group G is

V \simeq \bigoplus_{\lambda \in \Lambda} \langle V^\lambda,V\rangle V^\lambda,

where \langle V^\lambda,V\rangle is the multiplicity of V^\lambda in V.

Note that the multiplicity \langle V^\lambda,V\rangle is a nonnegative integer and not literally a scalar product. The reason to write multiplicities in this suggestive way is to bring out an analogy with the Fourier basis \{F^\lambda \colon \lambda \in \Lambda\} of the convolution algebra of \mathcal{C}(G) of an abelian group G, where the unitary dual \Lambda of G simultaneously parameterizes homomorphisms from G to the unit circle and elements of G itself. In the abelian case we have

A =\frac{1}{|G|}\sum\limits_{\lambda \in \Lambda} \langle \chi^\lambda,A\rangle F^\lambda,

making the Fourier decomposition of a function appear similar to the isotypic decomposition of a representation.

Theorem 21.2. Two unitary representations (V,\varphi) and (W,\psi) of G are isomorphic if and only if \chi^V = \chi^W.

Proof: We showed in Lecture 20 that isomorphic representations have the same character. For the converse, suppose \chi^V=\chi^W. Let

V \simeq \bigoplus\limits_{\lambda \in \Lambda} \langle V^\lambda,V\rangle V^\lambda\quad\text{and}\quad W \simeq \bigoplus\limits_{\lambda \in \Lambda} \langle V^\lambda,W\rangle V^\lambda

be their isotypic decompositions. Then by Theorem 21.1 we have

\chi^V =\sum\limits_{\lambda \in \Lambda} \langle \chi^\lambda,\chi^V\rangle \chi^\lambda\quad\text{and}\quad \chi^W = \sum\limits_{\lambda \in \Lambda} \langle \chi^\lambda,\chi^W\rangle \chi^\lambda,

and hence the hypothesis \chi^V=\chi^W implies

\sum\limits_{\lambda \in \Lambda} \langle \chi^\lambda,\chi^V\rangle \chi^\lambda = \sum\limits_{\lambda \in \Lambda} \langle \chi^\lambda,\chi^W\rangle \chi^\lambda.

Since \{\chi^\lambda \colon \lambda \in \Lambda\} is a linearly independent set in \mathcal{Z}(G), this forces

\langle \chi^\lambda,\chi^V\rangle = \langle \chi^\lambda,\chi^W\rangle, \quad \lambda \in \Lambda.

Thus, the isotypic decompositions of the representations (V,\varphi) and (W,\psi) are the same, and therefore these representations are isomorphic.

-QED

Theorem 21.3. A unitary representation (V,\varphi) of G is irreducible if and only if \langle \chi^V,\chi^V\rangle=|G|.

Proof: We have already proved one direction of this result: for any \lambda \in \Lambda, we have \langle \chi^\lambda,\chi^\lambda\rangle=|G|. For the converse, suppose it is the case that \langle \chi^V,\chi^V\rangle=|G|. Let

V \simeq \bigoplus_{\lambda \in \Lambda} \langle V^\lambda,V\rangle V^\lambda

be the isotypic decomposition of V, so that

\chi^V \simeq \bigoplus_{\lambda \in \Lambda} \langle V^\lambda,V\rangle \chi^\lambda.

We then have

\langle \chi^V,\chi^V\rangle = \sum\limits_{\lambda,\mu \in \Lambda} \overline{\langle V^\lambda,V\rangle}\langle V^\mu,V\rangle \langle \chi^\lambda,\chi^\mu\rangle=|G|\sum\limits_{\lambda\in \Lambda} |\langle V^\lambda,V\rangle|^2,

so the hypothesis \langle \chi^V,\chi^V\rangle=|G| forces

\sum\limits_{\lambda\in \Lambda} |\langle V^\lambda,V\rangle|^2=1.

This in turn means that the sum \sum\limits_{\lambda\in \Lambda} |\langle V^\lambda,V\rangle|^2 has exactly one term equal to one and all other terms equal to zero, so that the isotypic decomposition of V is V\simeq V^\lambda for some \lambda \in \Lambda.

-QED

As you may recall from some lectures back, every finite group G has a distinguished unitary representation, namely the regular representation (\mathcal{C}(G),\rho) in which \mathcal{C}(G) is viewed as a Hilbert space with the \ell^2-scalar product and

\varphi(g)A = E_gA, \quad g \in G.

Explicitly, this is the function on G given by

[E_gA](h) = \sum\limits_{k \in G}E_g(k)A(k^{-1}h) = A(g^{-1}h), \quad h \in G.

Another description of the regular representation is to say how \varphi(g) acts on the group basis of \mathcal{C}(G),

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

Problem 21.2. Compute the isotypic decomposition

\mathcal{C}(G) = \bigoplus\limits_{\lambda \in \Lambda} \langle V^\lambda,\mathcal{C}(G)\rangle V^\lambda

of the regular representation. It is quite important that you solve this problem, because we will use the result. To get started, compute the character of the regular representation.

Published by Jonathan Novak

Mathematician at UC San Diego.

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