Whether the division algebra of morphisms of a SO3 representation on symmetric trace zero 3 by 3 matrices , is Real, Complex or Quaternions

Question

Let $V$ be the space all symmetric matrices in $M_{3}(\mathbb{R})$ with trace zero. Consider the representation $\rho$ of $SO_{3}(\mathbb{R})$ on $V$ given by, $$ \rho : SO_{3}(\mathbb{R}) \to GL(V) \\ Q \mapsto (A \mapsto QAQ^{t}) $$ I got that this representation is irreducible, and by Schur's Lemma the algebra $End_{SO_{3}(\mathbb{R})}(V)$ (the algebra of endomorphisms of $V$ invarient under $\rho$) is a division algebra. Hence by Frobenius Theorem, it is isomorphic to either $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. How do find which one it is? I would prefer an algebraic proof of this.

Answer 1

The usual way to do this is to first extend the representation over reals to a representation over complex numbers and analyse what happens.

Let $V$ be a representation of a group $G$ over $\mathbb{R}$. The complexified space $V_{\mathbb{C}} = V \otimes \mathbb{C}$ over $\mathbb{C}$ is naturally a representation of $G$ over $\mathbb{C}$, and every equivariant linear map $F \colon V \to V$ extends to an equivariant linear map $F_{\mathbb{C}} \colon V_{\mathbb{C}} \to V_{\mathbb{C}}$ of complex spaces as $F_{\mathbb{C}} (x \otimes \alpha) = F(x) \otimes \alpha$. We have $$\operatorname{End}_G(V_{\mathbb{C}}) = [\operatorname{End}(V_{\mathbb{C}})]^G = [\operatorname{End}(V) \otimes \mathbb{C}]^G = \operatorname{End}_G(V) \otimes \mathbb{C}.$$

By Schur's lemma, over $\mathbb{C}$ every irrep has endomorphism ring isomorphic to $\mathbb{C}$. We have three cases:

• If $\operatorname{End}_G(V) \cong \mathbb{R}$, then $\operatorname{End}_G(V_{\mathbb{C}}) \cong \mathbb{C}$, which means that $V_{\mathbb{C}}$ is irreducible.
• If $\operatorname{End}_G(V) \cong \mathbb{C}$, then $\operatorname{End}_G(V_{\mathbb{C}}) \cong \mathbb{C} \otimes \mathbb{C} = \mathbb{C} \times \mathbb{C}$, which means that $V_{\mathbb{C}}$ is a direct sum of two non-isomorphic irreps over $\mathbb{C}$ (which in this case would be conjugates of each other)
• If $\operatorname{End}_G(V) = \mathbb{H}$, then $\operatorname{End}_G(V_{\mathbb{C}}) \cong \mathbb{H} \otimes \mathbb{C} = \mathbb{C}^{2 \times 2}$, which means that $V_{\mathbb{C}}$ is an irrep with multiplicity 2: $V_{\mathbb{C}} = W \oplus W$.

In our case, $V_{\mathbb{C}}$ is the complex vector space of traceless symmetric matrices, which is irreducible as a complex representation, so $V$ has endomorphism ring isomorphic to $\mathbb{R}$.