# Abstract Nonsense

## A Characterization of Real, Complex, and Quaternionic Irreps

Point of post: In this post we derive a result historically attributed to Frobenius and Schur which gives us a characterization to real, complex, and quaternionic irreps based on their admittant characters.

Motivation

In the past we’ve discussed how the set of all irreps are naturally carved up into three subclasses: real, complex, and quaternionic. This analogizes the difference between real, complex, and quaternionic numbers. It turns out that in general it is not, at first glance, clear how to determine from elementary methods whether or not an irrep was real, complex, or quaternionic. Indeed, in our one example of quaternionic irreps the agrument that the irrep in question was, in fact, quaternionic was involved and admittedly convoluted. That said, the theorem we develop in this post shall give us a simple way to determine whether an irrep is real, complex, and conjugate by a simple calculation involving the character of the irrep.

Theorem of Frobenius and Schur

Without further ado, we proceed to the main theorem:

Theorem: Let $G$ be a finite group and $\rho^{(\alpha)}:G\to\mathcal{U}\left(\mathscr{V}\right)$ be an irrep, with $\dim_{\mathbb{C}}\mathscr{V}=n$  and  induced character $\chi_\rho$. Then,

$\displaystyle \frac{1}{|G|}\sum_{g\in G}\chi^{(\alpha)}\left(g^2\right)=\begin{cases}1 & \mbox{if}\quad \rho^{(\alpha)}\text{ is real}\\ 0 & \mbox{if}\quad \rho^{(\alpha)}\text{ is complex}\\ -1 & \mbox{if}\quad \rho^{(\alpha)}\text{ is quaternionic}\end{cases}$

Proof: We note first that given any fixed matrix representation $D^{(\alpha)}$ for $\rho^{(\alpha)}$

\begin{aligned}\displaystyle \frac{1}{|G|}\sum_{g\in G}\chi^{(\alpha)}\left(g^2\right) &=\frac{1}{|G|}\sum_{g\in G}\sum_{j=1}^{n}D^{(\alpha)}_{j,j}\left(g^2\right)\\ &=\frac{1}{|G|}\sum_{g\in G}\sum_{j=1}^{n}\sum_{r=1}^{n}D^{(\alpha)}_{j,r}(g)D^{(\alpha)}_{r,j}(g)\\ &= \sum_{j=1}^{n}\sum_{r=1}^{n}\frac{1}{|G|}\sum_{g\in G}D^{(\alpha)}_{j,r}(g)D^{(\alpha)}_{r,j}(g)\end{aligned}\quad\quad\mathbf{(1)}

Suppose first then that $\rho$ is real, then there exists a basis $\mathcal{B}$ for $\mathscr{V}$ such that $\left[\rho(g)\right]_{\mathcal{B}}\in\text{Mat}_n\left(\mathbb{R}\right)$ for every $g\in G$ and so in particular if $latex K^{(\alpha)}$ is the associated matrix representation then $K^{(\alpha)}_{i,j}$ is a real function. Thus,  since $\mathbf{(1)}$ is true for any matrix representation it must be true with respect to the matrix representation associated with $\mathcal{B}$. But, we then have that $D^{(\alpha)}_{r,j}(g)=\overline{D^{(\alpha)}_{r,j}(g)}$ for each $g\in G$. Thus, the last step of $\mathbf{(1)}$ may, in light of the orthogonality of the matrix entry functions, as

$\displaystyle \sum_{j=1}^{n}\sum_{r=1}^{n}\delta_{j,r}\delta_{r,j}=1$

In agreeance with our desired result.

Suppose next that $\rho^{(\alpha)}$ is complex and let $J$ be any complex conjugate for $\mathscr{V}$. We know then from a previous characterization of complex conjugates that there exists some ordered orthonormal basis $\mathcal{B}$ for $\mathcal{B}$ such that $J(v)=v$ for each $v\in\mathcal{B}$. We can clearly see then that if $D^{(\alpha)}$ is the matrix realization of $\rho^{(\alpha)}$ with respect to this $\mathcal{B}$ and $W^{(\beta)}$ the realization of $\text{Conj}^J_{\rho^{(\alpha)}}$ with respect to $\mathcal{B}$ then for every $k,\ell\in d_\alpha$ and $g\in G$ $D^{(\alpha)}_{k,\ell}(g)=\overline{W^{(\beta)}_{k,\ell}(g)}$ where since $\rho^{(\alpha)}$ we may conclude that $\alpha\ne\beta$. It follows then that, once again, we may interpret this last part of $\mathbf{(1)}$ as the inner product of $D^{(\alpha)}_{j,r}$ with $W^{(\beta)}_{r,j}$ and since $\alpha\ne\beta$ this must be equal to zero in agreeance with the desired result.

Lastly, suppose that $\rho^{(\alpha)}$ is quaternionic. Since $\rho^{(\alpha)}$ is self conjugate for any complex conjugate $J$ on $\mathscr{V}$ one has that there exists unitary $U$ such that $\rho^{(\alpha)}(g)=U\text{Conj}^J_{\rho^{(\alpha)}}(g) U^{-1}$ for each $g\in G$ and since, using the same argument as before we can choose an ordered basis for which then

$\displaystyle D^{(\alpha)}_{k,\ell}(g)=\sum_{p,q=1}^{n}U_{k,p}\overline{D^{(\alpha)}_{p,q}(g)}\overline{U_{\ell,q}}$

Plugging this into the last step of $\mathbf{(1)}$ and applying the orthogonality relations we may conclude that the last step of $\mathbf{(1)}$ may be simplified to

$\displaystyle \frac{1}{|G|}\sum_{g\in G}D^{(\alpha)}_{j,r}(g)D^{(\alpha)}_{r,j}(g)=\frac{1}{n} U_{j,r}\overline{U_{r,j}}$

and so clearly

$\displaystyle \frac{1}{|G|}\sum_{g\in G}\chi\left(g^2\right)=\frac{1}{n}\text{tr}\left(U\overline{U}\right)$

Recalling though our characterization of self-conjugate irreps one can see that $U\overline{U}$ is similar to $-\mathbf{1}$ so that $\displaystyle \frac{1}{n}\text{tr}\left(U\overline{U}\right)=-1$ as desired. The conclusion follows. $\blacksquare$

Rerefences:

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.