Finite Arithmetics Applying to Certain Geometric Algebras

摘要

Integers 0 through 8, indexing the components of a vector in Euclidean eight-dimensional space E8, are represented in three-binary-digit notation. A no-carry arithmetic is introduced, in which the binary digits in each of the three binary decimal positions are summed modulo 2. The arithmetic permits arraying seven ordered triples, each composed of three of the seven non-zero component indices, in eight distinct though systematically related lists. Each list leads algorithmically to a 336-term form antisymmetric and quadrilinear in the components of four eight-vectors, a form invariant under a 21-parameter subgroup of the 28-parameter rotation group in E8. Each form implies a vector-cross-product of three eight-vectors. More directly, each list represents a 42-term antisymmetric form trilinear in components of three seven-vectors, invariant under a 14-parameter rotation subgroup implied in the list, and thence to a vector-cross-product of two seven-vectors. The lists lead to multiplication tables for Cayley-number units, and to classification of some related 8 X 8 signed arrays found or implied in the earlier literature.