Math 202B: Lecture 13

***All problems in this lecture due Feb. 15 at 23:59***

Let $G$ be a finite group. We have been comparing and contrasting the function algebra $\mathcal{F}(G)$ and the convolution algebra $\mathcal{C}(G)$ . The function algebra $\mathcal{F}(G)$ sees $G$ as a finite set, and does not interact with its group structure. The convolution algebra does interact with the group structure of $G$ , and in particular is commutative if and only if $G$ is abelian.

We understand subalgebras of $\mathcal{F}(G)$ very well: they are in bijection with partitions of $G$ . Namely, if $\mathfrak{p}$ is a partition of $G,$ then we have a corresponding subalgebra of $\mathcal{F}(G)$ , the set of functions on $G$ which are constant on the blocks of $\mathfrak{p}.$

Our understanding of subalgebras of $\mathcal{C}(G)$ is less complete. We know that every subgroup $H$ of $G$ gives rise to a corresponding subalgebra $\mathcal{A}(H)$ of $\mathcal{C}(G),$ namely the set of functions on $G$ which vanish outside $H.$ We also used the averaging trick to construct a two-dimensional subalgebra of $\mathcal{C}(G)$ of a different kind.

Since pointwise multiplication of functions and convolution of functions are very different operations, there is no reason to expect that $\mathcal{A}(\mathfrak{p})$ should be closed under convolution. But perhaps this could be the case for some “special” partition of $G$ which arises from its group structure.

Partitions of $G$ are the same thing as equivalence relations on $G.$ In the context of group theory, there is a particularly important equivalence relation: conjugacy. By definition, two points $g,h \in G$ are conjugate to one another if and only if $g = khk^{-1}$ for some $k \in G.$ This equivalence relation defines a partition of $G$ whose blocks are called conjugacy classes. In group theory, the significance of this concept is that a subgroup of $G$ is normal if and only if it is a union of conjugacy classes. We will now consider the partition of $G$ into conjugacy classes from the algebra perspective. Functions on $G$ which are constant on conjugacy classes are know as class functions. Could it be the case that the set of class functions on $G$ is a subalgbra of $G$ ?

Let $\{C_\alpha \colon \alpha \in \Lambda\}$ be the partition of $G$ into conjugacy classes. Here $\Lambda$ is a finite set of labels which parameterize the blocks of this partition.

Problem 13.1. Prove that $|\Lambda|=|G|$ if and only if $G$ is abelian, and that $|\Lambda|=1$ if and only if $|G|=1.$ Next, prove that the cardinality of the set $\{(g,h) \in G \times G \colon gh=hg\}$ of commuting pairs of elements from $G$ is $|G||\Lambda|.$

We have defined class functions to be those functions on $G$ which are constant on its conjugacy classes. A useful alternative characterization of such functions is that they are precisely those functions which are insensitive to any noncommutativity in $G.$

Problem 13.2. A function $A \in \mathcal{C}(G)$ is a class function if and only if $A(gh)=A(hg).$

Now we return to our original question: is the set of class functions on $G$ a subalgebra of $\mathcal{C}(G)$ ? The answer is yes, as the following theorem shows.

Theorem 13.1. The set of class functions on $G$ is precisely the center of $\mathcal{C}(G).$

Proof: Suppose first that $A \in \mathcal{C}(G)$ is a class function, and let $B \in \mathcal{C}(G)$ be any function. We want to prove that $A$ commutes with $B.$ According to the definition of convolution, we have

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

where to get the second equality we used the substation $h \rightsquigarrow gh^{-1}$ and to get the third inequality we used the commutativity of $\mathbb{C}.$ Since $A$ is a class function, we conclude that

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

Conversely, suppose that $A \in \mathcal{C}(g)$ is a central function on $G.$ Then, it commutes with each of the elementary functions $E_g$ , which form a basis of $\mathcal{C}(G)$ as $g$ ranges over $G$ . Now observe that

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

whereas

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

Since these are equal due to the centrality of $A$ , Problem 13.2 shows that $A$ is a class function. $\square$

The fact that the set of class functions is a subalgebra of $\mathcal{C}(G)$ now follows from the fact that the center of any algebra is a subalgebra thereof. Let us write $\mathcal{K}(G):=Z(\mathcal{C}(G))$ for brevity.

Theorem 13.2. The dimension of $\mathcal{K}(G)$ is equal to $|\Lambda|,$ the number of conjugacy classes in $G$ .

Proof: Let

$K_\alpha = \sum\limits_{g \in C_\alpha} E_g$

be the indicator function of the conjugacy class $C_\alpha.$ Then, the set $\{K_\alpha \colon \alpha \in \Lambda\}$ spans $\mathcal{K}(G).$ Moreover, for the $L^2$-scalar product on $\mathcal{C}(G)$ we have

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

Since $\{C_\alpha \colon \alpha \in \Lambda\}$ is a partition of $G,$ this gives

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

hence $\{K_\alpha \colon \alpha \in \Lambda\}$ is an orthogonal set of nonzero functions. $\square$

It is useful to think of $\mathcal{K}(G)$ and $\mathcal{C}(G)$ on the same footing, as two algebras associated to any finite group. The class algebra $\mathcal{K}(G)$ is always commutative, while the convolution algebra $\mathcal{C}(G)$ is commutative precisely when $G$ is abelian, and coincides with $\mathcal{K}(G)$ in this case. Any group $G$ with at least two elements has at least two conjugacy classes, hence the dimension of $\mathcal{K}(G)$ is at least two – the convolution algebra $\mathcal{C}(G)$ of a nontrivial group can be noncommutative, but not a noncommutative as the endomorphism algebra of a Hilbert space, whose center is one-dimensional.

An interesting aspect of the class algebra $\mathcal{K}(G)$ is that the connection coefficients of the class basis $\{K_\alpha \colon \alpha \in \Lambda\}$ count solutions to equations in $G.$ More precisely, consider the $|\Lambda|^3$ structure constants defined by