Abstract Nonsense

Irreducible R-characters of a Finite Group in Terms of its Irreducible C-characters

Point of post: In this post we combine our results over the last few posts to give a complete description of the irreducible $\mathbb{R}$-characters of a finite group $G$ in terms of its irreducible $\mathbb{C}$-characters.

Motivation

We’ve spent a lot of time in our last few posts finding a bijection between a subset (which we denoted by $\Delta_G$) equivalency classes of of $\mathbb{C}$-representations and irreducible $\mathbb{R}$-representations, finding a way to create a bunch of elements of $\Delta_G$, and showing a necessary condition for membership in $\Delta_G$. Thus, we’ve completely ascertained, in a sense, the identity of $\Delta_G$ and from this we shall show how to compute the irreducible $\mathbb{R}$-characters of $G$ from the irreducible $\mathbb{C}$-characters. From this we’ll derive the result that if $\widehat{G}_\Re$ denotes the equivalency classes of irreducible $\mathbb{R}$-representations then

$\displaystyle \sum_{\alpha\in\widehat{G}_\Re}d_\alpha^2\geqslant|G|$

with equality if and only if $\alpha$ is real for every $\alpha\in\widehat{G}$.

Characterization of Irreducible $\mathbb{R}$-characters

Let $G$ be a finite group, $\Delta_G$ the set of all equivalency classes of $\mathbb{C}$-representations $\rho$  of $G$ which satisfy the real condition and if their realizer is $J$ then the representation space of $\rho$ has no non-trivial proper $\left(\rho,J\right)$-invariant subspaces, and $\widehat{G}_\Re$ the set of all equivalency classes of irreducible $\mathbb{R}$-representations.

Theorem: If $\chi^{(\alpha_r)}$ is an irreducible $\mathbb{R}$-character of $G$ where $\alpha_r\in\widehat{G}_\Re$ then either

$\text{ }$

\begin{aligned}&\textbf{(1)}\quad \textit{There exists some real }\alpha\in\widehat{G}\textit{ such that }\chi^{(\alpha_r)}=\chi^{(\alpha)}\\ &\mathbf{(2)}\quad \textit{There exists some quaternionic }\alpha\in\widehat{G}\textit{ such that }\chi^{(\alpha_r)}=2\chi^{(\alpha)}(g)\\ &\textbf{(3)}\quad \textit{There exist some complex }\alpha\in\widehat{G}\textit{ such that }\chi^{(\alpha_r)}=\chi^{(\alpha)}+\chi^{(\overline{\alpha})}=2\text{Re}\left(\chi^{(\alpha)}\right)\end{aligned}

$\text{ }$

Moreover, every character of the above three forms appears in $\widehat{G}_\Re$.

Proof: We know from our result concerning the bijection of $\Delta_G$ with $\widehat{G}_\Re$ that there exists some $[\rho]\in\widehat{G}$ so that $\chi^{(\alpha_r)}=\text{tr}\left(\rho_\Re\right)$. But, we know that for each $[\rho]\in\Delta_G$ one has that $\rho$ is equivalent to $\psi$ where $\psi$ is some real irreducible $\mathbb{C}$-representation, $\psi\oplus\text{Conj}_\psi^J$ for some quaternionic irreducible $\mathbb{C}$-representation, or $\psi\oplus\text{Conj}_\psi^J$ for some complex irreducible $\mathbb{C}$-representation. But, since $\text{tr}(\psi)=\text{tr}(\psi_\Re)$ we see that these last three possibilities account for the three possibilities listed above. Moreover, since we know that each possibility is attainable the conclusion follows. $\blacksquare$

From this we get the two following fascinating corollaries:

Corollary:

\displaystyle \begin{aligned}\#\left(\widehat{G}_\Re\right) &=\#\left\{\text{conjugacy classes of }G\right\}-\frac{1}{2}\#\left\{\text{complex }\alpha\in\widehat{G}\right\}\\ &=\frac{1}{2}\left(\#\left\{\text{conjugacy classes of }G\right\}+\#\left\{\text{ambivalent conjugacy classes of }G\right\}\right)\end{aligned}

$\text{ }$

(where ambivalent conjugacy classes are defined as before)\

$\text{ }$

And

Corollary: Let $G$ be a finite group and for each $\alpha\in\widehat{G}_\Re$ let $d_\alpha$ denote its degree. Then,

$\text{ }$

$\displaystyle \sum_{\alpha\in\widehat{G}}d_\alpha^2=|G|+\sum_{\alpha\in\widehat{G}\text{ s.t. }\alpha\text{ is complex}}d_\alpha^2+3\sum_{\alpha\in\widehat{G}\text{ s.t. }\alpha\text{ is quaternionic}}d_\alpha^2$

$\text{ }$

And thus $\displaystyle \sum_{\alpha\in\widehat{G}}d_\alpha^2\geqslant |G|$ with equality if and only if every irrep of $G$ is real.

$\text{ }$

$\text{ }$

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References:

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

April 4, 2011 -