Half-spin representation ($S^{+},S^{-}$)

For $\mathrm{Spin}(2m)$, $m\ge 3$, the two complex half-spin irreducible representations have dimension $2^{m-1}$. Their weights are $\frac{1}{2}(\pm e_1\pm \cdots \pm e_m)$ with respectively even and odd numbers of minus signs, after a chirality convention is chosen; every weight multiplicity is one. A model is the even or odd part of the exterior algebra of ${\mathbb{C}}^m$, where an occupied mode reverses the corresponding sign. Each is a minuscule representation. These are representations of the spin group and its Lie algebra, but do not descend through the double covering to the special orthogonal group.