Spin representation

A spinor representation is a representation of a Spin group, or infinitesimally of a Special orthogonal Lie algebra, constructed from a Clifford algebra. In even dimensions it can split into two chiral spinor representations.

The Spin group $\mathrm{Spin}(7)$ double covers SO(7). Its central kernel element $-1$ acts trivially in the vector representation and as $-I$ in the spinor representation. Thus a spinor Lie algebra representation does not define an ordinary SO(7) group representation. Under $\mathrm{Spin}(6)\cong SU(4)$ the same obstruction is visible because $-I_4$ acts trivially on ${\Lambda }^{2}4$ but negatively on $4\oplus \overline{4}$.

Choose five Hermitian four-by-four matrices obeying the Euclidean Clifford algebra relations ${\gamma }_a{\gamma }_b+{\gamma }_b{\gamma }_a=2{\delta }_{ab}I$. For $M_{ab}=E_{ab}-E_{ba}$, the displayed anti-Hermitian matrices satisfy the Special orthogonal Lie algebra Lie bracket relations. A direct proof first establishes $[\rho (M_{ab}),{\gamma }_c]={\delta }_{bc}{\gamma }_a-{\delta }_{ac}{\gamma }_b$, then applies the commutator product rule to $[{\gamma }_c,{\gamma }_d]$. One explicit choice is ${\gamma }_1={\sigma }_1\otimes I$, ${\gamma }_2={\sigma }_2\otimes I$, ${\gamma }_3={\sigma }_3\otimes {\sigma }_1$, ${\gamma }_4={\sigma }_3\otimes {\sigma }_2$, ${\gamma }_5={\sigma }_3\otimes {\sigma }_3$, with the Pauli matrices. The Cartan subalgebra operators $-i\rho (M_{12})$ and $-i\rho (M_{34})$ have simultaneous eigenvalues $(\pm 1/2,\pm 1/2)$, identifying the four-dimensional spinor representation. A full $2\pi$ rotation acts as $-I_4$, proving it does not descend to $SO(5)$.

For the B2 root system with long simple root first, the four-dimensional spinor representation has highest weight $(0,1)$. Its symmetric square has highest weight $(0,2)$ and dimension ten, so it is the Adjoint representation. The exterior square has the four nonzero defining weights and two zero weights; it splits into the five-dimensional vector representation of highest weight $(1,0)$ and a trivial representation. Thus the tensor square decomposes as displayed. The invariant alternating bilinear form reflects the pseudoreal representation structure. The spinor representation is a group representation of the Spin group $\mathrm{Spin}(5)$, while its tensor square descends to $SO(5)$.

An irreducible complex Clifford algebra module in even spacetime dimension $2k$ has dimension $2^k$ and splits under the connected Spin group into two Weyl spinor modules of dimension $2^{k-1}$. In odd dimension $2k+1$, an irreducible spinor still has dimension $2^k$ but has no Weyl splitting. Reality conditions depend on signature and are additional to these complex counts.