Spinors from Pure Spinors

ABSTRACT We propose and develop a new method to classify orbits of the spin group Spin (2d) in the space of its semi-spinors. The idea is to consider spinors as being built as a linear combination of their pure constituents. As is well-known, two pure spinors can sum up to a pure spinor. This means that in a realization with the minimal number of pure spinors, no pair of pure spinor constituents can sum up to a pure spinor. Imposition of this constraint leads to a simple combinatorial problem that has a finite number of solutions in dimensions up to and including 14. A less well-known phenomenon occurs in dimension 12, where a sum of four pure spinors satisfying the pairwise constraint can actually be represented as a pair of pure spinors. Taking this phenomenon into account imposes constraints that we call tetrahedral. With these constraints added, the associated combinatorial problem has a finite number of solutions in dimensions up to and including 18, where no results are available. We call each distinct solution a combinatorial type of an impure spinor. We represent each combinatorial type graphically by a simplex, with vertices corresponding to the pure constituents of a spinor, and edges being labelled by the dimension of the totally null space that is the intersection of the annihilator subspaces of the pure spinors living at the vertices. Similarly, higher dimensional cells are labelled by the dimension of the common annihilator subspace of all pure spinors bordering the cell. We call the number of vertices in a simplex the impurity of an impure spinor. In dimensions 8 and 10, the maximal impurity is 2. Dimension 12 is the first dimension where one gets an impurity 3 spinor, represented by a triangle. In dimension 14, the generic orbit has impurity 4, while the maximal impurity is 5. We show that each of our combinatorial types uniquely corresponds to one of the known spinor orbits, thus reproducing the classification of spinors in dimensions up to and including 14 from simple combinatorics. Our methods continue to work in dimensions 16 and higher, but the number of the possible distinct combinatorial types grows rather rapidly with the dimension.