KAIST Discrete Math Seminar

In this talk, the diagrams of affine permutations and their balanced labellings will be introduced. 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 column strict balanced labellings is the affine Stanley symmetric function defined by Lam. The affine Stanley symmetric function is the object of active research in the field of Schubert calculus. It is the affine counterpart of the Stanley symmetric function which is the limit of Schubert polynomials. Our construction is a natural tableau-theoretic realization of this function. We also give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we will give a necessary and sufficient condition for a diagram to be an affine permutation diagram. If time allows, we will introduce some conjectures about when the affine Stanley symmetric functions coincide. This talk is based on the joint work with Taedong Yun.