Balanced labellings of affine permutations

Balanced labellings of affine permutations

Blog Article

We study the $ extit{diagrams}$ of affine permutations seashell peel and stick wallpaper and their $ extit{balanced}$ labellings.As in 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 $ extit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings.

Applying this theory, we st albans alpaca scarf give a necessary and sufficient condition for a diagram to be an affine permutation diagram.Finally, we conjecture that if two affine permutations are $ extit{diagram equivalent}$ then their affine Stanley symmetric functions coincide.