Balanced labellings of affine permutations

Balanced labellings of affine permutations

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We study the $ extit{diagrams}$ of affine permutations iphone xr price calgary 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 colts dog collar Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings.Applying this theory, we 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.