Half-spin representation

Definition

Let $V$ be an oriented complex quadratic space of even dimension $2m$ , and let $\Delta$ be an irreducible module for its complex Clifford algebra. The chirality operator decomposes the spinor module as

\Delta=\Delta^+\oplus\Delta^-.

The two eigenspaces $\Delta^+$ and $\Delta^-$ are invariant under $\operatorname{Spin}(V)$ and carry the two half-spin representations. Each has complex dimension $2^{m-1}$ , and Clifford multiplication by a vector interchanges them.

Highest weights

For $m\geq4$ , the complex Lie algebra $\mathfrak{so}_{2m}(\mathbb C)$ has Dynkin type $D_m$ . With a standard labeling of its two spin nodes, the half-spin modules have highest weights

\omega_{m-1}\quad\text{and}\quad\omega_m.

They are therefore the two spin fundamental representations. Which weight is called $+$ and which is called $-$ depends on the orientation, the labeling of the $D_m$ diagram, and the normalization of the Clifford volume element.

The central element $-1\in\operatorname{Spin}(2m)$ acts nontrivially on spinors. Consequently, half-spin representations are genuine representations of the spin group and do not descend to $SO(2m)$ .

Low-dimensional examples

The accidental isomorphism $\operatorname{Spin}(4)\cong SU(2)\times SU(2)$ identifies $\Delta^+$ and $\Delta^-$ with the defining two-dimensional representations of the two respective factors. Under $\operatorname{Spin}(6)\cong SU(4)$ , the two half-spin representations become the defining four-dimensional representation and its dual. In dimension eight, the two half-spin representations and the vector representation all have dimension eight and participate in triality.

Chirality and real forms

The complex decomposition into $\Delta^+$ and $\Delta^-$ is intrinsic only after the relevant orientation and chirality conventions are fixed; reversing orientation exchanges the labels. For a real group $\operatorname{Spin}(p,q)$ , whether either complex half-spin module admits an invariant real or quaternionic structure depends on $p-q$ modulo eight. A statement about “real half-spinors” must therefore specify the signature.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989, Chapter I, §§4–5. Publisher record.
William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, §20. Publisher record.