The Connection Between the Square Root Function and the Number of Self-Conjugate Irreps (Cont.)
Point of post: This post is a continuation of this one.
Ambivalent Conjugacy Classes
Let be a finite group and let be a conjugacy class in . We say that is ambivalent if where is the inversion map . Since and consequently is finite and is a bijection, it follows from first principles that being ambivalent is equivalent to . Now, for notational convenience for we denote by and so ambivalence of takes either of the two equivalent forms or . We now show the relationship between ambivalent conjugacy classes of a finite group and the number of self-conjugate . Indeed:
Theorem: Let be a finite group with conjugacy classes and let and be the number of ambivalent conjugacy classes in and the number of self-conjugate respectively. Then, .
Proof: Note first that
Indeed, note that if is any complex conjugate of any irrep then it’s clear that for every (this follows by considering our earlier characterization of complex conjugate maps) and since it follows that the left-hand side of can be considered as where denotes the element of containing from where the claim follows. Clearly then from this and the fact that each is a class function we see that
But, by definition is equal to if and only if is conjugate to which is true if and only if which is true if and only if . Thus, is one if is ambivalent and zero otherwise. So,
from where the conclusion follows.
1. Isaacs, I. Martin. Character Theory of Finite Groups. New York: Academic, 1976. Print.
2. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.