The Evolution of Coxeter-Dynkin Diagrams

Abstract

‘Dynkin diagrams’ or ‘Coxeter-Dynkin diagrams’ are graphs whose vertices and edges represent generators and relations (respectively) for so-called Coxeter groups. The generators have period two, and each of the relations specifies the period of the product of two generators. For instance, the group of automorphisms of the 27 lines on the general cubic surface (the Wey1 group E 6) is generated by six involutions, each of which interchanges the two rows of a ‘double six’. An early precursor of the appropriate diagram (see page 4) was used in 1904 by C. Rodenberg to indicate how six double-sixes are related. The same diagrams arise in various connections; for instance, the vertices may represent the mirrors of a kaleidoscope while the edges indicate the angles between pairs of mirrors.