A simple Lie group $(8)$ has 2 half-spinor representations and 1 vector representation (coming from standard vector representation of SO(8)), all of them have dimension 8. One can compose a representation of a Lie group, with an automorphism of the group to get another representation. In $(8)$ case, $((8))$ permutes above three representations and this we get an isomorphism of $((8))$ and $_3$. $(8)$ is the most symmetrical one in the sense that $((8))=S_3$ is the largest possible group.
My question concerns what are the matrix matrix representations of the 28 Lie algebra generators for the 2 half-spinor representations and 1 vector representation.
- The vector representation of $Spin(8)$ is also the vector representation of $SO(8)$. We know the 27 of rank-8 matrices can be obtained by taking any two basis vectors $v_i$ and $v_j$ of 8-dimensional vector space, and assign $\pm 1$ along the off diagonal component with this matrix: $$\begin{bmatrix} 0&-1\\1&0\end{bmatrix}$$ namely along the wedge product of the two basis vectors: $$v_i \wedge v_j = v_i \otimes v_j - v_j \otimes v_i.$$ Include this matrix into the rank-8 matrix, we get $\frac{8 \cdot 7}{2}=28$ such set of matrix $$\begin{bmatrix} 0& \cdots & 0 & 0 \\ 0& \cdots &-1 &\vdots\\ \vdots & 1& \ddots & \vdots \\ 0& \cdots & \cdots & 0 \end{bmatrix}$$ namely along the wedge product of the two basis vectors: $$v_i \wedge v_j = v_i \otimes v_j - v_j \otimes v_i.$$ for any $i,j\in \{1,2,\dots,7,8\}$ with $i \neq j$.
Above we derive the explicit Lie algebra matrix representations of $Spin(8)$ also $SO(8)$: vector representation, with 28 Lie algebra generators of rank-8.
- Question --- How do we derive the explicit Lie algebra matrix representations of $Spin(8)$: Two half-spinor representations? with 28 Lie algebra generators of rank-8?