In this talk we will introduce some combinatorial objects motivated by the
Schubert calculus of the affine Grassmannian.
We give "affine" generalizations of the notions of semistandard Young
tableaux and the RSK (Robinson-Schensted Knuth) algorithm. Our affine
tableaux come in two versions, a weak and a strong one. Weak and strong
tableaux are certain chains in the weak and strong (Bruhat) orders of the
affine symmetric group respectively.
Our main result is an affine insertion algorithm, generalizing the RSK
algorithm, which sends matrices to pairs (P,Q) of a strong tableau P and a
weak tableau Q of the same shape. As a special case, we find that the
weak order and strong order (labeled by affine Chevalley coefficients) of the affine symmetric group forms a
pair of dual graded graphs in the sense of Fomin.
While our work is geometrically motivated, no knowledge of geometry will
be assumed. This is joint work with Lapointe, Morse and Shimozono