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 J. Y. Shi (J. London Math. Soc.35 (1987), 42–55). As applications, we give an affirmative answer to a conjecture of H. Eriksson and K. Eriksson (preprint, 1998) concerning the characterization of the inverse table of w Wa; we also describe the number πs(w) in terms of the permutation form of w Wa.