# Abstract Nonsense

## The Connection Between the Square Root Function and the Number of Self-Conjugate Irreps

Point of post: In this post we find a formula for the number of $\alpha\in\widehat{G}$ such that $\rho^{(\alpha)}$ is self-conjugate for any $\alpha\in\widehat{G}$. As a by-product we must define the notion of an ambiguous conjugacy class in a group, and derive a relationship between the number of ambiguous subsets and the square root function.

Motivation

We have seen, using our last post, that there is an intimate relation between the square root function $\sqrt{\text{ }}:G\to\mathbb{N}\cup\{0\}$ and the notion of self-conjugate representation. Indeed, we have seen that an irrep $\rho^{(\alpha)}:G\to\mathcal{U}\left(\mathscr{V}\right)$ is self-conjugate if and only if $\displaystyle \frac{1}{|G|}\sum_{g\in G}\sqrt{g}\chi^{(\alpha)}(g)=\pm 1$. In this post we show that the relation deepens by showing that one can explicitly calculate the number of $\alpha\in\widehat{G}$ such that $\rho^{(\alpha)}$ is a self-conjugate irrep for any $\alpha\in\widehat{G}$  in terms of $\sqrt{\text{ }}$. Incidentally, we shall show there is another way to count the number of self-conjugate $\alpha$ of a group by matching them up with the number of ‘ambivalent’ conjugacy classes in a group. Ambivalent conjugacy classes got their namesake via their apparent ambivalent nature as to whether they want to be subgroups in the sense that they are closed under inversion but not necessarily under multiplication. In connecting ambivalent subsets and the number of self-conjugate irreps we will inevitably show a relationship between the number of ambivalent subsets of the group and the square root function.

The Number of Self-Conjugate Irreps of a Finite Group

To save notation, we recall that being real, complex, or quaterinonic as an irrep is invariant under equivalence (althought I actually neglected to prove this when I first discussed real, complex, and quaternionic irreps since equivalent irreps admit the same irreducible character this follows immediately from our recent characterization). Thus, for any $\alpha\in\widehat{G}$ one has that the adjectives  real, complex, quaternionic, and self-conjugate apply simultaneously to every element of $\alpha$.. Thus, it makes sense to define $\alpha$ to be real, complex, quaternionic, or self-conjugate  if any one of its elements is real, complex,quaternionic, or self-conjugate  respectively. We define then, in accordance with previous definitions, the function $c:\widehat{G}\to\{-1,0,1\}$ with $c(\alpha)$ denoted $c_\alpha$ to be $1,0$ and $-1$ if $\alpha$ is real, complex, or quaternionic respectively. With this in mind we proceed to derive our first result. Namely:

Theorem: Let $G$ be a finite group and $\mathfrak{s}$ the number of self-conjugate $\alpha\in\widehat{G}$. Then,

$\displaystyle \mathfrak{s}=\frac{1}{|G|}\sum_{g\in G}\sqrt{g}\;^2$

Proof: Recall from a previous result that

$\displaystyle \sqrt{g}=\sum_{\alpha\in\widehat{G}}c_\alpha\chi^{(\alpha)}(g)$

Consequently,

\displaystyle \begin{aligned}\frac{1}{|G|}\sum_{g\in G}\sqrt{g}\;^2 &= \frac{1}{|G|}\sum_{g\in G}\sqrt{g}\;\overline{\sqrt{g}}\\ &= \frac{1}{|G|}\sum_{g\in G}\sum_{\alpha\in\widehat{G}}\sum_{\beta\in\widehat{G}}c_\alpha c_\beta \chi^{(\alpha)}(g)\overline{\chi^{(\beta)}(g)}\\ &= \sum_{\alpha,\beta\in\widehat{G}}c_\alpha c_\beta\frac{1}{|G|}\sum_{g\in G}\chi^{(\alpha)}(g)\overline{\chi^{(\beta)}(g)}\\ &= \sum_{\alpha,\beta\in\widehat{G}}c_\alpha c_\beta \delta_{\alpha,\beta}\\ &= \sum_{\alpha\in\widehat{G}}c_\alpha^2\\ &=\mathfrak{s}\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.