Let $W$ be a real irreducible representation of finite group $G$. According to the Frobenius Theorem on real division algebra, $\text{End}_G(W)$ is isomorphic to either the field of real numbers, the field of complex numbers, or the division ring of quaternions.

I want to find an isomorphism in each case.

I have some ideas, which I wrote in the following. Please tell me whether they are correct and help me to find an isomorphism for the last case.

Let $\tau \in \operatorname{End}_G(W)$.
Case 1 ($\operatorname{End}_G(W) \cong \mathbb{R}$): By schure lemma, $\tau=\lambda \cdot 1_W$. So, $\lambda=\dfrac{\operatorname{trace}(\tau)}{\dim(W)}$. Consequently, $$\tau \mapsto \dfrac{\operatorname{trace}(\tau)}{\dim(W)}$$ is an isomorphism.

Case 2 ($\operatorname{End}_G(W) \cong \mathbb{C}$): I suspect $\tau$ has the matrix form $$ \begin{bmatrix}\lambda_1& \lambda_2\\-\lambda_2 & \lambda_1 \end{bmatrix} \otimes I_{\frac{1}{2}\dim(W)}. $$ So, \begin{align} \lambda_1 &= \frac{\operatorname{trace}(\tau)}{\dim(W)} \\[4pt] \lambda_2 &= \sqrt{{\operatorname{det}(\tau)}^{\frac{n}{2}} - \biggl(\frac{\operatorname{trace}(\tau)}{\dim(W)}\biggr)^2}, \end{align} Consequently, $$\tau \mapsto \dfrac{\operatorname{trace}(\tau)}{\dim(W)}+i \sqrt{{\operatorname{det}(\tau)}^{\frac{n}{2}} - \biggl(\frac{\operatorname{trace}(\tau)}{\dim(W)}\biggr)^2}$$ is an isomorphism.

Case 3 ($\operatorname{End}_G(W) \cong \mathbb{H}$): No idea



You must log in to answer this question.

Browse other questions tagged .