Abstract Nonsense

Crushing one theorem at a time

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.



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:



\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{ }




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{ }

\text{ }


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


April 4, 2011 - Posted by | Algebra, Representation Theory | , , ,

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