# Finite Geometry, Dirac Groups and the Table of Real Clifford Algebras

Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 321)

## 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 quotient group V n = G(p, q)/ *#x007B;± 1*#x007D;, viewed additively, is an ndimensional vector space over GF(2) = *#x007B;0, 1*#x007D; which comes equipped with a quadratic form Q and associated alternating bilinear form B. Properties of the finite geometry over GF(2) of V n B, Q — in part familiar, in part less so — are given a rather full description, and a dictionary of translation into their Dirac group counterparts is provided. The knowledge gained is used, in conjunction with facts concerning representations of G(p, q), to give a pleasantly clean derivation of the well-known table of Porteous (1969/1981) of the algebras Cl(p, q). In particular the finite geometry highlights the “antisymmetry” of the table about the column p - q = -1. Several low-dimensional illustrations are given of the application of finite geometry results to Dirac groups. Particular emphasis is laid on certain interesting finite geometry symmetry methods, the latter being given a rather full treatment in the appendices. Finite geometry is also used to study the automorphisms of the Dirac groups, and the splitting of certain exact sequences.

## Key words

Finite geometry Clifford algebras Dirac groups Caps on quadrics Conwell heptads Non-polar sets Symmetric groups Split exact sequences

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## Authors and Affiliations

• R. Shaw
• 1
1. 1.School of MathematicsUniversity of HullHullEngland