On Two Presentations of the Affine Weyl Groups of Classical Types

The main result of the paper is to get the transition formulae between the alcove form and the permutation form of w ∈ Wa, where Wa is an affine Weyl group of classical type. On the other hand, we get a new characterization for the alcove form of an affine Weyl group element which has a much simpler form compared with that in [10]. As applications, we give an affirmative answer to a conjecture of H. Eriksson and K. Eriksson in [4] concerning the characterization of the inverse table of w ∈ Wa; we also describe the number πs(w) in terms of permutation form of w ∈ Wa. Introduction. Affine Weyl groups, as a family of infinite crystallographic Coxeter groups, play a more and more important role in various fields of mathematics, such as Kac-Moody algebras, algebraic groups and their representation theory, combinatorial and geometric group theory, etc. [2; 3; 5; 7; 9; 14]. Besides the presentations as Coxeter groups (i.e., the ones by generators and relations over the pairs of generators), there are many other presentations for the affine Weyl groups Wa, in particular for those of the classical types, i.e., types Ãl (l > 1), B̃m (m > 3), C̃n (n > 2) and D̃k (k > 4). Two of their presentations are particularly useful, one is to regard Wa as a certain permutation group over the integer set Z (only applied for the classical types); the other is to identify Wa with the set of alcoves in a euclidean space E after removing a certain set of hyperplanes (applied for all types) [1; 8; 9; 10; 11; 12; 13].