The study of Coxeter groups in general could be said to first arise in the classification of regular polyhedra (the Platonic solids) in ancient Greece. The modern systematic study (connecting the algebraic and geometric definitions of finite and affine Coxeter groups) began in work of Coxeter in the 1930s. The combinatorial description of the affine symmetric group first appears in work of Lusztig (1983), and was expanded upon by Shi (1986) ; both authors used the combinatorial description to study the Kazhdan–Lusztig cells of S ~ ~ n . The proof that the combinatorial definition agrees with the algebraic definition was given by Eriksson & Eriksson (1998).