The positive half-spin representation of Spin(8)

The positive half-spin representation $8_+=\Delta^+$ of compact $\operatorname{Spin}(8)$ is its action on one of the two irreducible real modules of the even Clifford algebra

\mathrm{Cl}^{0}_{0,8}\cong M_8(\mathbb R)\oplus M_8(\mathbb R).

Here Clifford generators satisfy $v^2=-\lVert v\rVert^2$ . For an oriented orthonormal basis, put $\omega=e_1\cdots e_8$ . The positive module is the simple module on which $\omega$ acts by $+1$ . The module has real dimension $8$ , and restriction to $\operatorname{Spin}(8)$ is irreducible.

Central kernel

The action has image $SO(8)$ after choosing an oriented orthonormal basis. Its kernel is an order-two central subgroup different from both the kernel of the vector representation and that of the negative half-spin representation. In particular, it does not descend through the fixed vector quotient $\operatorname{Spin}(8)\to SO(8)$ .

Chirality convention

The labels $+$ and $-$ depend on orientation; reversing orientation exchanges them. This catalogue keeps both objects. Triality changes the group action when it exchanges them.

References

John C. Baez, “The Octonions,” §2.4, Table 4 and restriction from the even Clifford algebra. Checked section.
Schaposnik and Schulz, “Triality for Homogeneous Polynomials,” §2.2, p. 5. Paper.