Abstract
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.
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