They are named for Richard Brauer and Hermann Weyl,[1] and were one of the earliest systematic constructions of spinors from a representation theoretic standpoint.
Let n = 2k (or 2k+1) and suppose that the Euclidean quadratic form on V is given by where (pi, qi) are the standard coordinates on Rn.
It is no longer important to distinguish between the Ps and Qs, so we shall simply refer to them all with the symbol P, and regard the index on Pi as ranging from i = 1 to i = 2k.
By counting dimensions, A is a complete 2k×2k matrix algebra over the complex numbers.
Consider the application of an orthogonal transformation to the coordinates, which in turn acts upon the Pi via That is,
By decomposing rotations into products of reflections, one can write down a formula for S(R) in much the same way as in the case of three dimensions.
The ambiguity defines S(R) up to a nonevanescent scalar factor c. Since S(R) and cS(R) define the same transformation (1), the action of the orthogonal group on spinors is not single-valued, but instead descends to an action on the projective space associated to the space of spinors.
However, there is only one irreducible pin representation (see below) owing to the non-invariance of the above eigenspace decomposition under improper rotations, and that has dimension 2k.
In the quantization for an odd number 2k+1 of dimensions, the matrices Pi may be introduced as above for i = 1,2,...,2k, and the following matrix may be adjoined to the system: so that the Clifford relations still hold.
It may further be extended to general orthogonal transformations by setting S(R) = -S(-R) in case det R = -1 (i.e., if R is a reversal).
This fact is not evident from the Weyl's quantization approach, however, and is more easily seen by considering the representations of the full Clifford algebra.