The Center of a Character (Pt. I)
Point of post: In this post we discuss the notion the ‘center of a character’ and its relations to other concepts we’ve previously learned.
In previous posts we’ve seen how we can take concepts from plain group theory and attempt to make an analogy for characters. In this post we extend this further by defining the ‘center’ of a character, which, as we shall see are just those for which . In other words, its just those for which the modulus of is maximized. This shall prove useful in the future, in particular it shall help us devolop enough machinery to prove Burnside’s Theorem–the classic use of representation theory.
Center of a Character
Let be a finite group and some character of . Then, we define the center of , denoted , to be
Our first claim is that, despite what is not immediately apparent, for every . Indeed, this follows immediately from:
Theorem: Let be a finite group and the character of the representation . Then,
Proof: It’s clear by the unitarity of for every that if then . Thus, and so as desired.
Conversely, suppose that . Recall from the spectral theorem that we have that is similar to where and are the eigenvalues of with multiplicity. But, by assumption we have that
from where it follows from the fact that one must have that and so is similar to and thus . The conclusion follows.
Corollary:Let be a finite group and a character for . Then, .
Proof: This clearly follows since if then and and so evidently and so . The conclusion follows.
If we have a representation of a finite group one can easily verify that for any subgroup of the restriction of to is a representation of . Moreover, the character of restricted to is also a character on . Also, it’s apparent that if is irreducible then is irreducible. Thus, if for some then for some . With this in mind, we make the following claim:
Theorem: Let be a finite group and a character of . Then, for some degree-one irrep .
Proof: We know by definition that for every there exists some for which . Define by . This is clearly a degree-one irrep and moreover it’s clear that . Thus,
from where the conclusion follows.
1. Isaacs, I. Martin. Character Theory of Finite Groups. New York: Academic, 1976. Print.