Math 202B: Lecture 25

Let \Lambda be a set parameterizing irreducible unitary representations of G up to isomorphism, and for each \lambda \in \Lambda let (V^\lambda,\varphi^\lambda) be a representative of the corresponding isomorphism class. Equivalently, \Lambda parameterizes irreducible linear representations of the convolution algebra \mathcal{C}(G) up to isomorphism, and for each \lambda \in \Lambda we take the representative (V^\lambda,\Phi^\lambda) determined by

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

Thus, for any function

A = \sum\limits_{g \in G} A(g) E_g

in \mathcal{C}(G) we have

\Phi^\lambda(A) = \sum\limits_{g \in G} A(g)\phi^\lambda(g),

and

\chi^\lambda(A) = \mathrm{Tr}\ \Phi^\lambda(A) = \sum\limits_{g \in G} A(g) \mathrm{Tr}\, \varphi^\lambda(g) = \sum\limits_{g \in G} A(g) \chi^\lambda(g).

Recall that the class algebra \mathcal{K}(G) consists of those functions A \in \mathcal{C}(G) which are constant on the conjugacy classes of G, and that this is the center of \mathcal{C}(G). By Schur’s Lemma, if A \in \mathcal{K}(G) then the operator \Phi^\lambda(A) is a scalar multiple of the the identity operator,

\Phi^\lambda(A) = \omega^\lambda(A)I_{V^\lambda}.

This determines a map

\omega^\lambda \colon \mathcal{K}(G) \longrightarrow \mathbb{C}

called the central character of the irrep (V^\lambda,\Phi^\lambda).

Problem 25.1. Prove that, for any \lambda \in \Lambda, the central character \omega^\lambda \colon \mathcal{K}(G) \to \mathbb{C} is an algebra homomorphism, given in terms of the character of (V^\lambda,\Phi^\lambda) by

\omega^\lambda(A) = \frac{1}{\dim V^\lambda} \chi^\lambda(A), \quad A \in \mathcal{K}(G).

We are now ready to prove that the irreducible characters of G form an orthogonal basis of the class algebra \mathcal{K}(G). We

Theorem 25.2. The set \{\chi^\lambda \colon \lambda \in \Lambda\} spans \mathcal{K}(G).

Proof: Let W be the subspace of \mathcal{K}(G) spanned by \{\chi^\lambda \colon \lambda \in \Lambda\}. We will show that W^\perp consists solely of the zero function. That is, if A \in \mathcal{K}(G) is a central function orthogonal to every irreducible character,

\langle A, \chi^\lambda \rangle = \sum\limits_{g \in G} \overline{A(g)}\chi^\lambda(g) =0, \quad \lambda \in \Lambda,

then A is the zero function on G. Let

\overline{A}=\sum\limits_{g \in G} \overline{A(g)}E_g,

which is the conjugate of A viewed as an element of the function algebra \mathcal{F}(G) rather than the convolution algebra \mathcal{C}(G). Then,

\omega^\lambda(\overline{A})= \frac{1}{\dim V^\lambda} \mathrm{Tr} \Phi^\lambda(\overline{A}) = \frac{1}{\dim V^\lambda} \sum\limits_{g \in G} \overline{A(g)} \chi^\lambda(g) \frac{1}{\dim V^\lambda} \langle A,\chi^\lambda\rangle= 0, \quad \lambda \in \Lambda.

This says that the image of \overline{A} in every irreducible representation (V^\lambda,\Phi^\lambda) of \mathcal{C}(G) is the zero operator. Since every linear representation of \mathcal{C}(G) is a direct sum of irreducible linear representations, this means that \overline{A} is the zero operator in every linear representation of \mathcal{C}(G). This includes the regular representation, which is faithful. We conclude that \overline{A} and hence A is the zero function on G, as required. \square

We now know that the set \{\chi^\lambda \colon \lambda \in \Lambda\} of irreducible characters of G is an orthogonal basis of \mathcal{K}(G). In particular, |\Lambda|=\mathrm{dim}\mathcal{K}(G), which says the following.

Corollary 25.3. The number of isomorphism classes of irreducible unitary representations of G is equal to the number of conjugacy classes in G.

Let us consider character orthogonal more carefully. Explicitly, we have

\langle \chi^\lambda,\chi^\mu\rangle= \sum\limits_{g \in G} \overline{\chi^\lambda(g)}\chi^\mu(g) = \sum\limits_{g \in G}\chi^\lambda(g^{-1})\chi^\mu(g)=\delta_{\lambda\mu}|G|.

Now, let \{C_\alpha \colon \alpha \in \Lambda\} be an enumeration of the conjugacy classes in G, and let us write \chi^\lambda_\alpha for \chi^\lambda(g) with g \in C_\alpha. Then, character orthogonality takes the form

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

Definition 25.4. The character table of G is the square matrix \mathbf{T} with rows and columns indexed by \Lambda and entries

\mathbf{T}_{\alpha\lambda} = \chi^\lambda_\alpha.

The modified character table of G is the square matrix \mathbf{M} with rows and columns indexed by \Lambda and entries

\mathbf{M}_{\alpha\lambda} = \sqrt{\frac{|C_\alpha|}{|G|}}\chi^\lambda_\alpha.

Character orthogonality says that the modified character table of G is a unitary matrix, i.e. that columns of \mathbf{M} are pairwise orthogonal unit vectors,

\sum\limits_{\alpha \in \Lambda} \overline{\mathbf{M}}_{\lambda\alpha} \mathbf{M}_{\mu\alpha}=\sum\limits_{\alpha \in \lambda} \sqrt{\frac{|C_\alpha|}{|G|}}\overline{\chi^\lambda_\alpha}\sqrt{\frac{|C_\alpha|}{|G|}}\chi^\mu_\alpha=\delta_{\lambda\mu}.

Since the rows of a unitary matrix are also pairwise orthogonal unit vectors, this gives us a second orthogonality relation, namely

\sum\limits_{\lambda \in \Lambda} \overline{\mathbf{M}}_{\lambda\alpha} \mathbf{M}_{\lambda\beta}=\sum\limits_{\lambda \in \lambda} \sqrt{\frac{|C_\alpha|}{|G|}}\overline{\chi^\lambda_\alpha}\sqrt{\frac{|C_\beta|}{|G|}}\chi^\lambda_\beta=\delta_{\alpha\beta}.

This is called dual character orthogonality.

Theorem 25.5. For any \alpha,\beta \in \Lambda, we have

\sum\limits_{\lambda \in \Lambda} \overline{\chi^\lambda_\alpha}\chi^\lambda_\beta=\delta_{\alpha\beta}\frac{|G|}{|C_\alpha|}.

As an application of Dual Character Orthogonality, we can obtain a formula for the connection coefficients of the class algebra \mathcal{K}(G) in terms of the irreducible characters of G. Recall that the class basis

K_\alpha=\sum\limits_{g \in C_\alpha} E_g, \quad \alpha \in \Lambda

of \mathcal{K}(G) has the property that its connection coefficients

K_\alpha K_\beta = \sum\limits_{\gamma \in \Lambda} c_{\alpha\beta\gamma}K_\gamma, \quad \alpha,\beta \in \Lambda

solve a factorization problem in the group G.

Problem 25.2. Prove that

c_{\alpha\beta\gamma} = \frac{|C_\alpha||C_\beta|}{|G|}\sum\limits_{\lambda \in \lambda} \frac{\chi^\lambda_\alpha \chi^\lambda_\beta \overline{\chi^\lambda_\gamma}}{\dim V^\lambda}.

Published by Jonathan Novak

Mathematician at UC San Diego.

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