# Abstract Nonsense

## 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 $G$ be a finite group and let $\mathcal{C}$ be a conjugacy class in $G$. We say that $\mathcal{C}$ is ambivalent if $\iota\left(\mathcal{C}\right)\subseteq\mathcal{C}$ where $\iota:G\to G$ is the inversion map $g\mapsto g^{-1}$. Since $G$ and consequently $\mathcal{C}$ is finite and $\iota:G\to G$ is a bijection, it follows from first principles that being ambivalent is equivalent to $\mathcal{C}=\iota\left(\mathcal{C}\right)$. Now, for notational convenience for $A\subseteq G$ we denote $\iota\left(A\right)$ by $A^{-1}$ and so ambivalence of $\mathcal{C}$ takes either of the two equivalent forms $\mathcal{C}^{-1}\subseteq\mathcal{C}$ or $\mathcal{C}^{-1}=\mathcal{C}$. We now show the relationship between ambivalent conjugacy classes of a finite group $G$ and the number of self-conjugate $\alpha\in\widehat{G}$. Indeed:

Theorem: Let $G$ be a finite group  with conjugacy classes $\mathcal{C}_1,\cdots,\mathcal{C}_k$ and let $\mathfrak{a}$ and $\mathfrak{s}$ be the number of ambivalent conjugacy classes in $G$ and the number of self-conjugate $\alpha\in\widehat{G}$ respectively. Then, $\mathfrak{a}=\mathfrak{s}$.

Proof: Note first that

$\displaystyle \frac{1}{|G|}\sum_{g\in G}\chi^{(\alpha)}(g)^2=|c_\alpha|\quad\quad\mathbf{(1)}$

Indeed, note that if $\text{Conj}^J_{\rho^{(\alpha)}}$ is any complex conjugate of any irrep $\rho^{(\alpha)}\in\alpha$ then it’s clear that $\chi_{\text{Conj}^J_{\rho^{(\alpha)}}}(g)=\overline{\chi^{(\alpha)}(g)}$ for every $g\in G$ (this follows by considering our earlier characterization of complex conjugate maps) and since $\chi^{(\alpha)}(g)=\overline{\overline{\chi^{(\alpha)}(g)}}$ it follows that the left-hand side of $\mathbf{(1)}$ can be considered as $\left\langle \chi^{(\overline{\alpha})},\chi^{(\alpha)}\right\rangle$ where $\overline{\alpha}$ denotes the element of $\widehat{G}$ containing $\text{Conj}^J_{\rho^{(\alpha)}}$ from where the claim follows. Clearly then from this and the fact that each $\chi^{(\alpha)}$  is a class function we see that

\displaystyle \begin{aligned}\mathfrak{s} &= \frac{1}{|G|}\sum_{\alpha\in\widehat{G}}\sum_{g\in G}\chi^{(\alpha)}(g)^2\\ &=\sum_{j=1}^{k}\frac{\#\left(\mathcal{C}_j\right)}{|G|}\sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}\left(g_j\right)^2\end{aligned}

where $g_j$ is any element of $\mathcal{C}_j$. Recall though that $\chi^{(\alpha)}\left(g_j\right)=\overline{\chi^{(\alpha)}\left(g_j^{-1}\right)}$. Thus, it follows from the second orthogonality relation that

$\displaystyle \sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}(g_j)^2=\frac{|G|}{\#\left(\mathcal{C}_j\right)}c\left(g_j,g_j^{-1}\right)$

But, by definition $c\left(g_j,g_j^{-1}\right)$ is equal to $1$ if and only if $g_j$ is conjugate to $g_j^{-1}$ which is true if and only if $\mathcal{C}_j=\mathcal{C}_{g_j^{-1}}$ which is true if and only if $\mathcal{C}_j=\mathcal{C}_j^{-1}$. Thus, $c\left(g_j,g_j^{-1}\right)$ is one if $\mathcal{C}_j$ is ambivalent and zero otherwise. So,

\displaystyle \begin{aligned}\mathfrak{s} &= \sum_{j=1}^{k}\frac{\#\left(\mathcal{C}_j\right)}{|G|}\sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}\left(g_j\right)^2\\ &= \sum_{j=1}^{k}\frac{\#\left(\mathcal{C}_j\right)}{|G|}\frac{|G|}{\#\left(\mathcal{C}_j\right)}c\left(g_j,g_j^{-1}\right)\\ &= \sum_{j=1}^{k}c\left(g_j,g_j^{-1}\right)\\ &= \sum_{\mathcal{C}_j\text{ is ambivalent}}1\\ &=\mathfrak{a}\end{aligned}

from where the conclusion follows. $\blacksquare$

References:

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.