Finite Geometry and the Table of Real Clifford Algebras

Abstract

Associated with the real Clifford algebra Cl(p, q), p + q = n, is the finite Dirac group G(p,q) of order 2ⁿ⁺¹. The group V _n =G(p, q)/±1, viewed additively, is an n-dimensional vector space over GF(2) = 0, 1} which comes equipped with a quadratic form Q and associated alternating bilinear form B. The finite geometry of V _n , B, Q,in part familiar, in part less so, is described, and is then used in conjunction with the representation theory of G(p, q) to give a pleasantly clean derivation of the well-known table, [9], of the algebras Cl(p, q). In particular the finite geometry highlights the “antisymmetry” of the table about the column q - p = 1.