Ane Weyl groups as innite permutations

We present a unied theory for permutation models of all the innite families of nite and ane Weyl groups, including interpretations of the length function and the weak order. We also give new combinatorial proofs of Bott’s formula (in the rened version of Macdonald) for the Poincar es eries of these ane Weyl groups. 1991 Mathematics Subject Classication. primary 20B35; secondary 05A15.