Higher-dimensional extensions of Pauli spin matrices

The principle of basis set representation in terms of coordinate interchange matrices, of which the Pauli spin matrices are an example in two dimensions, are extended to three and four dimensions. The four-dimensional basis set of coordinate interchange matrices satisfies the usual conditions of completeness, but the three-dimensional basis set cannot be complete under any circumstances and an 'anticomplete' property is assigned to it. The coefficients of the basis set, when used to represent an arbitrary matrix, form a Hadamard transform of the cyclically interchanged arbitrary matrix. (author)