= Half-spin representation
{title2=$S^+,S^-$}
For $\operatorname{Spin}(2m)$, $m\ge3$, the two complex half-spin <irreducible representations> have dimension $2^{m-1}$. Their <weights> are $\tfrac12(\pm e_1\pm\cdots\pm e_m)$ 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>.
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