Intertwiners between the three Spin(8) modules

Statement

For compact $G=\operatorname{Spin}(8)$ , let $V,W$ range over its vector and two real half-spin modules. With the group action fixed,

\operatorname{Hom}_G(V,W)=\{0\}\quad(V\not\cong W), \qquad \operatorname{End}_G(V)=\mathbb R I, \qquad \operatorname{Aut}_G(V)=\mathbb R^\times I.

Here $\{0\}$ contains the zero map; it is not an empty Hom-set. Forgetting the action instead gives, after bases are chosen,

\operatorname{Hom}_{\mathbb R}(V,W)\cong M_8(\mathbb R), \qquad \operatorname{Aut}_{\mathbb R}(V)\cong GL(8,\mathbb R).

Direct proof

The three central kernels are distinct. Choose a central element acting as $+I$ on $V$ and $-I$ on a different $W$ . Equivariance forces $T=-T$ , hence $T=0$ .

Each action has image $SO(8)$ . If $T$ commutes with this image, the stabilizer of a unit vector $v$ forces $T(v)$ onto its line. Transitivity on the unit sphere then gives a single scalar $\lambda$ with $T=\lambda I$ . Such a map is invertible exactly when $\lambda\ne0$ .

Catalogue comparison

Each representation has a view in the category of real $G$ -modules and a view in real vector spaces. The categorical view chooses which equations a morphism must satisfy; it does not change the underlying eight-dimensional carrier.

References

Laura P. Schaposnik and Sebastian Schulz, “Triality for Homogeneous Polynomials,” §2.2, Proposition 2.1 and Remark 2.2, p. 5. Paper.