Math 202B: Lecture 8

The convolution algebra $\mathcal{C}(G)$ of a group $G$ is commutative if and only if $G$ is abelian. As in Lecture 2, we will refine this dichotomy by giving a quantitative measurement of how (non)commutative $\mathcal{C}(G)$ is. In Lecture 2 we defined the commutativity index of an algebra to be the dimension of its center, so our goal now is to determine the dimension of the center of a convolution algebra.

The degree of non commutativity of $\mathcal{C}(G)$ is determined by that of $G$ , and in group theory we understand this using the action of $G$ on itself defined by

$g.h = ghg^{-1}.$

The orbits of this action are the conjugacy classes of $G$ , and the number of conjugacy classes in $G$ is called its class number. The higher the class number, the more commutative the group – for abelian groups, conjugacy classes are singleton sets.

Problem 8.1. Prove that $G$ is an abelian group if and only if its class number is equal to its cardinality, and more generally that the number of commuting pairs of elements in $G$ is equal to the cardinality of $G$ times its class number (hint: the orbit-stabilizer theorem may be helpful). Show also that every group except the trivial group contains at least two conjugacy classes.

For the function algebra $\mathcal{F}(G)$ , every partition of $G$ gives rise to a subalgebra of $\mathcal{F}(G)$ , namely the set of functions constant on the blocks of the partition. As we discussed in Lecture 4, there is no reason that the convolution of two such functions will still be constant on the blocks of the given partition. On the other hand, the partition of $G$ into conjugacy classes is determined by the group structure of $G$ , and so functions on $G$ which are constant on the blocks of this particular group-theoretic partition of $G$ are relevant from the point of view of $\mathcal{C}(G).$

Definition 8.2. A function $A \colon G \to \mathbb{C}$ is called a class function if it is constant on conjugacy classes, meaning that $A(g) = A(hgh^{-1})$ for all $g,h \in G.$

The following gives an alternative way to think about class functions as functions which are insensitive to any noncommutativity present in $G$ .

Theorem 8.1. A function $A \colon G \to \mathbb{C}$ is a class function if and only if $A(gh)=A(hg)$ for all $g,h \in G.$

Proof: Suppose first that $A$ is a class function on $G$ . Then,

$A(gh) = A(hghh^{-1}) = A(hg).$

Conversely, suppose $A$ is insensitive to noncommutativity. Then,

$A(hgh^{-1}) = A(h^{-1}hg) = A(g).$

-QED

We can now characterize the center of $\mathcal{C}(G)$ .

Theorem 8.2. A function $A \in \mathcal{C}(G)$ belongs to $Z(\mathcal{C}(G))$ if and only if it is constant on conjugacy classes.

Proof: Suppose first that $A \in \mathcal{C}(G)$ is a class function; we will prove that it commutes with every $B \in \mathcal{C}(G).$ For any $g \in G$ , we have

$[AB](g) = \sum\limits_{h \in G} A(gh^{-1})B(h) = \sum\limits_{h \in G} A(g(gh)^{-1}) B(gh) = \sum\limits_{h \in G} A(gh^{-1}g^{-1})B(gh),$

where the second inequality follows from the fact that the substitution $h \rightsquigarrow gh$ simply permutes the terms of the sum. Continuing the calculation, we have

$[AB](g) = \sum\limits_{h \in G} A(gh^{-1}g^{-1})B(gh) =\sum\limits_{h \in G} A(h^{-1})B(gh) = \sum\limits_{h \in G} B(gh)A(h^{-1}),$

where the second inequality follows from the fact that $A$ is a class function. We now conclude

$[AB](g) = \sum\limits_{h \in G} B(gh^{-1})A(h) = [BA](g),$

as required.

Now suppose that $Z \in Z(\mathcal{C}(g))$ is a central function; we will prove it is constant on conjugacy classes. Since $Z$ commutes with all functions in $\mathcal{C}(g),$ it commutes with every elementary function $E_g$ . Since

$[ZE_g](h) = \sum\limits_{k \in G}Z(hk^{-1})E_g(k)=Z(hg^{-1}),$

and

$[E_gZ](h) = \sum\limits_{k\in G}E_g(hk^{-1})Z(k)=Z(g^{-1}h),$

the centrality of $Z$ implies that $Z(gh)=Z(hg)$ for all $g,h \in G$ . Thus $Z$ is a class function, by Theorem 5.1.

-QED

Thus, the set of all functions constant on conjugacy classes of $G$ is a subalgebra of both $\mathcal{F}(G)$ and $\mathcal{C}(G)$ . The subalgebra of class functions has no special significance in $\mathcal{F}(G)$ , not being any more or less special than the subalgebra associated with any other partition of $G$ . But in $\mathcal{C}(G)$ , the subalgebra of class functions is the center of $\mathcal{C}(G)$ . The center of $\mathcal{C}(G)$ is often called the class algebra of $G$ and denoted $\mathcal{Z}(G)$ rather than $Z(\mathcal{C}(G))$ to emphasize that it is worth thinking about as a standalone object, i.e. as a commutative algebra naturally associated to the finite group $G$ .

Just as the convolution algebra $\mathcal{C}(G)$ has a natural basis given by indicator functions of elements of $G,$ the class algebra $\mathcal{Z}(G)$ has a natural basis given by indicator functions of conjugacy classes in $G.$ Let $\Lambda$ be a set parameterizing the conjugacy classes of $G,$ so the collection of these is

$\{C_\alpha \colon \alpha \in \Lambda\},$

and this is a subset of the power set of $G.$ For each $\alpha \in \Lambda,$ let $K_\alpha \colon G \to \mathbb{C}$ be the indicator function of $C_\alpha,$ so

$K_\alpha(g) = \begin{cases} 1, \text{ if }g \in C_\alpha \\ 0, \text{ if }g \not\in C_\alpha \end{cases}.$

Equivalently, in terms of the group basis $\{E_g \colon g \in G\}$ of $\mathcal{C}(G)$ we have

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

Then, the functions $\{K_\alpha \colon \alpha \in \Lambda\}$ span the class algebra of $G,$ since any function $Z \colon G \to \mathbb{C}$ constant on conjugacy classes can be written

$Z = \sum\limits_{\alpha \in \Lambda} Z(\alpha) K_\alpha,$

where $Z(\alpha)$ denotes the value $Z(g)$ for any $g \in \mathcal{C}_\alpha.$ We will show that $\{K_\alpha \colon \alpha \in \Lambda\}$ is a basis of $\mathcal{Z}(G)$ using the $\ell^2$ -scalar product on $\mathcal{C}(G),$ which is

$\langle A,B \rangle = \sum\limits_{g \in G} \overline{A(g)}B(g).$

It is clear that $\{E_g \colon g \in G\}$ is an orthonormal basis of $\mathcal{C}(G)$ with respect to the $\ell^2$ -scalar product, and we therefore have

$\langle K_\alpha,K_\beta \rangle = \left\langle \sum\limits_{g \in C_\alpha} E_g , \sum\limits_{h \in C_\beta} E_h \right\rangle = \sum\limits_{g \in C_\alpha,\ h \in C_\beta} \langle E_g,E_h\rangle = |C_\alpha \cap C_\beta|.$

Since $\{C_\alpha \colon \alpha \in \Lambda\}$ is a partition of $G$ , the scalar product is

$\langle K_\alpha,K_\beta \rangle = \delta_{\alpha\beta}|C_\alpha|,$

which shows that $\{K_\gamma \colon \gamma \in \Gamma\}$ is an orthogonal (but not orthonormal) set of functions in $\mathcal{Z}(G)$ , hence linearly independent.

In conclusion, if $G$ is any finite group, the dimension of the convolution algebra $\mathcal{C}(G)$ is the cardinality of $G$ , and the dimension of the class algebra $\mathcal{Z}(G)$ is the class number of $G$ .

Problem 8.3. Consider the multiplication tensor $[c_{\alpha\beta\gamma}]$ of $\mathcal{Z}(G),$ i.e. $K_\alpha K_\beta = \sum_\gamma c_{\alpha\beta\gamma}K_\gamma.$ For any $\alpha,\beta,\gamma \in \Gamma$ and any $g \in C_\gamma$ , show that

$c_{\alpha\beta\gamma} = |\{(x,y) \in C_\alpha \times C_\beta \colon xy=g\}|.$

Thus, the multiplication tensor of $\mathcal{Z}(G)$ is quite an interesting object: $c_{\alpha\beta\gamma}$ counts solutions to the equation $xy=g$ in $G$ , where $g$ is any particular point of the conjugacy class $C_\gamma,$ and $x,y \in G$ are required to belong to $C_\alpha$ and $C_\beta$ respectively.

2 Comments

Srikiran Poreddy says:

January 19, 2025 at 1:27 pm

I am confused on problem 5.3. The right hand side is a set but the left hand side should be a number, right?

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1. Jonathan Novak says:
  
  January 19, 2025 at 7:19 pm
  
  Corrected, thanks.
  
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Math 202B: Lecture 8

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