A Generalization of the Robinson-Schensted-Knuth Correspondence

The Robinson-Schensted-Knuth (RSK) correspondence is a bijection between two-rowed lexicographic arrays and semistandard bitableaux. We will describe a generalization of this correspondence. For any Grassmannian permutation v, we will define a bijection between two-rowed lexicographic arrays and v-semistandard bitableaux, where v-semistandard bitableaux are generalizations of semistandard bitableaux whose shapes and contents depend on v. As an application, we will show how this generalization of the RSK correspondence can be used to obtain local properties at the T-fixed point [v] of a Schubert or Richardson variety in the Grassmannian.