Math 202B: Lecture 20

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Let (V,\varphi) and (W,\psi) be unitary representations of a finite group G. Let \chi^V,\chi^W\in \mathcal{C}(G) be their characters, i.e. the functions G \to \mathbb{C} defined by

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

Since \chi^V and \chi^W are defined using the trace, they are central functions on G: we have

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

and similarly \chi^W(gh)=\chi^W(hg). Thus, \chi^V and \chi^W belong to the class algebra \mathcal{Z}(G), which as you will recall is the center of the convolution algebra \mathcal{C}(G), or equivalently the subalgebra of \mathcal{C}(G) spanned by functions on G which are constant on the conjugacy classes of G.

Furthermore, observe that if (V,\varphi) and (W,\psi) are isomorphic, then their characters are equal: we have \chi^V(g)=\chi^W(g). Indeed, if these representations are isomorphic, then (by definition) there is a unitary vector space isomorphism T \colon V \to W which has the additional property that the equation

T\varphi(g) = \psi(g)T

holds in \mathrm{Hom}(V,W), for every g \in G. Equivalently, this is \psi(g) = T\varphi(g)T^*, whence

\chi^W(g) = \mathrm{Tr}\, \psi(g) = \mathrm{Tr}\, T\varphi(g)T^* = \mathrm{Tr}\, \varphi(g)T^*T = \mathrm{Tr}\, \varphi(g) = \chi^V(g).

Now let \Lambda be a set parameterizing isomorphism classes of irreducible unitary representations of G. At present we know nothing about \Lambda, and in particular we do not know whether or not it is finite. For each \lambda \in \Lambda, let (V^\lambda,\rho^\lambda) be a representative of the isomorphism class of irreducible unitary representations of G labeled by \lambda, and write \chi^\lambda for the character of this representation. Thus, we have a possibly infinite family of central functions on G,

\chi^\lambda, \quad \lambda \in \Lambda,

which are in bijection with isomorphism classes of irreducible unitary representations of G.

Theorem 20.1. \{\chi^\lambda \colon \lambda \in \Lambda\} is an orthogonal set in \mathcal{Z}(G).

Proof: For any \lambda,\mu \in \Lambda, the corresponding scalar product is

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

In Lecture 19, we proved that this scalar product is

\langle \chi^\lambda,\chi^\mu\rangle = |G| \dim \mathrm{Hom}_G(V,W).

Since (V^\lambda,\rho^\lambda) and (V^\mu,\rho^\mu) are irreducible representations, Schur’s Lemma tells us that

\dim \mathrm{Hom}_G(V^\lambda,V^\mu) = \delta_{\lambda\mu}.

We conclude that

\langle \chi^\lambda,\chi^\mu \rangle = \delta_{\lambda\mu}|G|.

-QED

Theorem 20.1 has many consequences. The first of these is that a finite group G has finitely many irreducible representations, up to isomorphism.

Theorem 20.2. The number of isomorphism classes of irreducible unitary representations of G is at most the class number of G.

Proof: Recall that the class number of G is the number of conjugacy classes in G, or equivalently the dimension of the class algebra \mathcal{Z}(G). According to Theorem 20.1, the set \{\chi^\lambda \colon \lambda \in \Lambda\} is orthogonal in \mathcal{Z}(G), therefore it is linearly independent and its cardinality is bounded by the dimension of \mathcal{Z}(G).

-QED

We have now established the bound |\Lambda| \leq \dim \mathcal{Z}(G). We will see next lecture that in fact equality holds: up to isomorphism, the number of irreducible unitary representations of G is equal to the number of conjugacy classes in G.

Published by Jonathan Novak

Mathematician at UC San Diego.

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