Balanced labellings of affine permutations
We study the $ extit{diagrams}$ of affine permutations and their $ extit{balanced}$ labellings.As in Knife Roll the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations.In fact, we show that the sum of weight monomials of the $ e