## Class Functions

**Point of post: **In this post we derive results about the set of class functions on a finite group , in particular finding its dimension as a subspace of the group algebra and characterizing it as the center of the group algebra.

**Motivation**

In our last series of posts we saw an interesting technique. We saw the interesting idea that if we want to prove the cardinality of a set is equal to it suffices to construct a vector space of dimension such that is a basis for . In particular, we saw that by considering the group algebra of dimension and then showing that is a basis for (where, as in the last post, the are the matrix entry functions). We now wish to get our milage out of this technique by applying it again in a different context (different set and different cardinality). We don’t want to ruin the surprise of what precisely this will be, but we shall construct the vector space with the ‘proper dimension’ in this post. In particular, we will consider and study the set of *class functions *on a finite group . Intuitively, these are functions which satisfy (of course we mean this loosely since we haven’t define the domain, range, etc.) the common ‘trace identity’ .

**Class Functions**

Let be a finite group. Then, if is a member of the group algebra which satisfies for every then we call a *class function*. We denote the set of all class functions by . We now derive some fairly elementary characterizations of the class functions.

**Theorem: ***Let be a finite group. Then for some the following are equivalent:*

* *

*(for those who need a reminder conjugacy classes and centers of algebras are defined here and here respectively)*

**Proof: **Although inefficient we prove this by and for maximum clarity.

: Suppose first that is a class function. Then, one has that for any . Conversely, if for every then for any one has that and so .

: Suppose first is constant on conjugacy classes then clearly since for any one has that is, by definition, conjugate to and thus in the same conjugacy classes. Thus, by assumption since were arbitrary the conclusion follows. Conversely, suppose that and be in the same conjugacy class. Then, by definition there exists some such that and so . Since in the conjugacy class of were arbitrary it follows that is constant on the conjugacy class of . Since was arbitrary it follows that is constant on all conjugacy classes of .

This is by far the trickiest one. Suppose first that then we note that for any one has that . That said, and from where the conclusion follows by the arbitrariness of . Conversely, suppose that for every then for any and any one has that

and since was arbitrary it follows that and since was arbitrary it follows that .

**Corollary: *** is a subalgebra of *

**Proof: **This follows from a previous theorem regarding the center of an algebra.

*Remark: *Note that this gives an alternate proof to our previous one that the group algebra is a commutative algebra if and only if is abelian. Indeed, is a commutative algebra if and only if which (with a little justification is true) if and only if is constant on each conjugacy class for each which is true if and only if eachc conjugacy class has one element which is true if and only if is abelian.

**References:**

1.Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. *Linear Representations of Finite Groups*. New York: Springer-Verlag, 1977. Print

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